Browse Trigonometry problems Studdy Solved!

Problem 1201

6) Convert $-65.2^{\circ}$ to radian measure.
Round your answer to the nearest hundredth of a radian.

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Problem 1202

If $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ , find $\tan \theta$ in surd form.

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Problem 1203

If $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ , find $\tan \theta$ in surd form.

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Problem 1204

Solve $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ for $\tan \theta$ in surd form.

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Problem 1205

Solve $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ for $\tan \theta$ in surd form.

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Problem 1206

Omar needs to find $\cos \left(-\frac{2 \pi}{3}\right)$ . Use the reference angle and quadrant info to solve.

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Problem 1207

Problem 1. Professor Naehrig is running on a circular track in a uniform circular motion. Her position can be described through the parametric equations $x=10+100 \cos \left(\frac{2}{3} \pi t+\frac{\pi}{4}\right) \text { and } y=100 \sin \left(\frac{2}{3} \pi t+\frac{\pi}{4}\right) \text {, }$ where $t$ is measured in minutes, the angle is measured in radians, and lengths are measured in meters.
1. What are the circle center coordinates?
2. Explain the meaning of $\frac{2}{3} \pi$ in the parametric equations.
3. Explain the meaning of $\frac{\pi}{4}$ in the parametric equations.
4. What is the radius of the circular track?
5. In coordinate system, sketch the circular track and Professor Naehrig's position at $t=0, t=0.5, t=1.5, t=3$ minutes.
6. Find the first 3 times when Professor Naehrig's $x$ -coordinate is $50 \sqrt{3}+10$ .
7. At the first time you found in the previous part, how far is she from the point $(20,30)$ ?
8. Find the first 3 times when Professor Naehrig's $y$ -coordinate is $50 \sqrt{2}$ . What is her $x$ -position at those respective times?

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Problem 1208

Find the sine, cosine, and tangent of $\angle K$ .
Simplify your answers and write them as proper fractions, improper fractions, or whole numbers. $\begin{array}{l} \sin (K)= \\ \cos (K)= \\ \tan (K)= \end{array}$ $\square$ $\square$ $\square$

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Problem 1209

A kite is flying 86 ft off the ground, and its string is pulled taut. The angle of elevation of the kite is $46^{\circ}$ . Find the length of the string. Round your answer to the nearest tenth. $\square$ ft

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Problem 1210

Suppose $y=3 \sin (4(t+13))-6$ . In your answers, enter pi for $\pi$ . (a) The midline of the graph is the line with equation $y=-6$ help (equations) (b) The amplitude of the graph is 3 help (numbers) (c) The period of the graph is $\pi$ help (numbers)
Note: You can earn partial credit on this problem.

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Problem 1211

example If $\operatorname{cosec} \theta=\sqrt{2}$ where $\theta$ is an acute angle, find (a) $\sec \theta(b) \tan \theta$ suppose $\cot \theta=\frac{3}{4}$ , where $\theta$ is an acute angle, find (a) $\operatorname{cosec} \theta$ (b) $\cos \theta$

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Problem 1212

Eliminate $\theta$ from the following equation (i). $x=\tan \theta+\cot \theta$ and $y=\tan \theta-\cot \theta$

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Problem 1213

Example (2) (Activity)
The range of the function $y=4 \sin 2 \theta$ is $\qquad$ (3) $[-4,4]$ (b) $\{-4,4\}$ (C) $[-2,2]$ (d) $]-2,2$ [ Sol.

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Problem 1214

Use the power reducing formulas to rewrite $\sin ^{4} x$ in terms of the first power of cosine. Simplify your answer as much as possible: To indicate your answer, first choose one of the four forms below. Then fill in the blanks with the appropriace numbers. $\sin ^{4} x=$ $\square$ $\square$ $\square$ $x+$ $\square$ $\square$ x $\sin ^{4} x=$ $\square$ $+$ $\square$ $\square \cos$ $\square$ ～ $x+$ $\square$ $\cos$ $\square$ $7 x$ $\sin ^{4} x=$ $\square$ - $\square$ $\cos$ $\square$ I $x$ $\sin ^{4} x=$ $\square$ $+$ $\square$ $\cos$ $\square$ x

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Problem 1215

11. Geben Sie die in kartesischer Binomialform gegebenen Punkte in Polarform an $A(-6 /-8), B=\left[8 ; 120^{\circ}\right], C(-5 / 3), D(5 / 0), E=\left[\sqrt{ } 12 ; 4,2^{\mathrm{rad}}\right], F=[5 ; \pi / 2]$
12. Skizzieren Sie am Einheitskreis die folgenden Funktionswerte. a. $\sin \left(250^{\circ}\right)$ b. $\cos \left(40^{\circ}\right)$ c. $\sin \left(\pi / 3^{\text {rad }}\right)$ d. $\cos \left(5^{\text {rad }}\right)$ e. $\sin \left(-70^{\circ}\right)$

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Problem 1216

Q14) The value of $4 \sin ^{-1}\left(\frac{\sqrt{5}}{19}\right)+4 \cos ^{-1}\left(\frac{\sqrt{5}}{19}\right)=$ a) $\pi$ b) $\frac{1}{2} \pi$ c) $\frac{1}{4} \pi$ d) $2 \pi$

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Problem 1217

Question 5
Consider a triangle $A B C$ like the one below. Suppose that $A=127^{\circ}, b=37$ , and $c=22$ . (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.
If there is more than one solution, use the button labeled "or".

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Problem 1218

Calculate $\frac{1}{\sin ^{2} x}$

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Problem 1219

$\tan x \cos x=0,0 \leq x \leq 2 \pi$

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Problem 1220

5. Given the expression $2 \cot (x)-2 \cot (x) \cos ^{2}(x)$ , a. Use technology to graph the expression [3 marks] b. Determine an equivalent trigonometric expression [2 marks] c. Then prove that your expression is equal to the given expression. [3 marks]

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Problem 1221

11. Provide a trigonometric equation. Considering only the space between $x=0$ and $2 \pi$ , the equation must only have solutions at $x=1$ and $x=2$ . Explain your thought process and the work you did to create the equation. You may round decimal values to 3 places. [ 6 marks]

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Problem 1222

For Exercises 25-30, assume that $\theta$ is an acute angle. (See Example 2)
25. If $\cos \theta=\frac{\sqrt{21}}{7}$ , find $\csc \theta$ .
26. If $\sin \theta=\frac{\sqrt{17}}{17}$ , find $\cot \theta$ .
27. If $\sec \theta=\frac{3}{2}$ , find $\sin \theta$ .
28. If $\csc \theta=3$ , find $\cos \theta$ .
29. If $\tan \theta=\frac{\sqrt{15}}{9}$ , find $\cos \theta$ .
30. If $\cot \theta=\frac{\sqrt{3}}{2}$ , find $\cos \theta$ .
Objective 3: Determine Trigonometric Function Values for Speçal Angles For Exercise 31, use the isosceles right triangle and the $30^{\circ}-60^{\circ}-90^{\circ}$ triangle to complete the table. (See Examples 3-4) 31. \begin{tabular}{|c|c|c|c|c|c|c|} \hline $\theta$ & $\sin \theta$ & $\cos \theta$ & $\tan \theta$ & $\csc \theta$ & $\sec \theta$ & $\cot \theta$ \\ \hline $30^{\circ}=\frac{\pi}{6}$ & & & & & & \\ \hline $45^{\circ}=\frac{\pi}{4}$ & & & & & $-\vdots$ \\ \hline $60^{\circ}=\frac{\pi}{3}$ & & & & & & \\ \hline \end{tabular}
32. a. Evaluate $\sin 60^{\circ}$ . b. Evaluate $\sin 30^{\circ}+\sin 30^{\circ}$ .

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Problem 1223

nvestigate.
1. Determine approximate solutions for each equation in the interval $x \in[0,2 \pi]$ , to the nearest hundredth of a radian. a) $\sin x-\frac{1}{4}=0$ b) $\cos x+0.75=0$ c) $\tan x-5=0$ d) $\sec x-4=0$ e) $3 \cot x+2=0$ f) $2 \csc x+5=0$

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Problem 1224

Example. $A$ sinusoidal function has an amplitude of 2 units, a period of $180^{\circ}$ and a maximum at $(0,3)$ . Write a possible equation for this function.

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Problem 1225

Q3: $\operatorname{For} z \in \mathbb{C}$ , show that: (a) $\sin \bar{z}=\overline{\sin z}$ ; (b) $\cosh \bar{z}=\overline{\cosh z}$ .

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Problem 1226

The period of is
Select one:

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Problem 1227

Determine the number of triangles $A B C$ possible with the given parts. $A=42.8^{\circ} \quad a=8.9 \quad b=10.1$
How many possible solutions does this triangle have? $\square$

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Problem 1228

\begin{problem} A plane flying at an altitude of 4 miles travels on a path directly over a radar tower.
(a) Express the distance $d(t)$ (in miles) between the plane and the tower as a function of the angle $t$ in standard position from the tower to the plane.
$d(t) = \square \csc \square \square \sin [$ \end{problem}

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Problem 1229

Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=$ $\square$

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Problem 1230

Verify that the equation is an identity. Show that $\csc x \cdot \frac{\tan x}{\cos x}=\sec ^{2} x$

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Problem 1231

Use a sum-to-product formula to find the exact value. Write your answer as a simplified fraction and rationalize the denominator, if nece $\sin 165^{\circ}-\sin 75^{\circ}=$ $\square$ $\sqrt{\text { 延 }}$

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Problem 1232

2 . In Exercises 23-28, graph three periods of the function. Use your understanding of transformations, not your grapher. Be sure to show the scale on both axes.
23. $y=5 \sin 2 x$
24. $y=3 \cos \frac{x}{2}$
25. $y=0.5 \cos 3 x$
26. $y=20 \sin 4 x$

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Problem 1233

Solve for $x$ in the equation $a \sec x = 3$ .

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Problem 1234

Solve for $x$ if $\sec x=3$ .

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Problem 1235

Prove that $\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec x$ .

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Problem 1236

Prove that $\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} x$ .

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Problem 1237

Show that $\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x$ .

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Problem 1238

Prove that $\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \theta$ .

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Problem 1239

Prove that $\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}$ .

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Problem 1240

Prove that $\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x$ .

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Problem 1241

Solve $9 \cot ^{2} \frac{1}{2} x=4$ for $-180^{\circ} \leq x \leq 180^{\circ}$ .

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Problem 1242

Solve $2 \operatorname{cosec}(2 x-1)=3$ for $-\pi \leq x \leq \pi$ .

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Problem 1243

Determine the other trigonometric functions for $\theta$ given that $\tan \theta=-\frac{1}{6}$ and $\sin \theta>0$ .

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Problem 1244

Find the exact value of $\tan \left(-\frac{\pi}{3}\right)$ using reference angles.

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Problem 1245

Find the exact value of $\sin \left(-300^{\circ}\right)$ using reference angles.

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Problem 1246

Find the exact value of $\cos \left(-\frac{\pi}{3}\right)$ using reference angles.

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Problem 1247

In $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

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Problem 1248

A 15 ft sliding board is at a $40^{\circ}$ angle. Find the height to the nearest tenth.

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Problem 1249

$\sin ^{-1}(1.94)=$ $\square$ $\tan ^{-1}(2.27)=$ $\square$ $\cos ^{-1}(-0.62)=$ $\square$

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Problem 1250

\begin{align} &\text{Given the function:} \\ &y = 3 \sin 2(x-1) + 3 \end{align} $\text{Why is its ending point between } 2\pi \text{ and } \frac{5\pi}{2}? \text{ What is that value and how were we supposed to know it ends there?}$

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Problem 1251

$\begin{array}{c}\tan ^{-1}(-1.53)= \\ \cos ^{-1}(2.26)= \\ \sin ^{-1}(0.42)=\end{array}$

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Problem 1252

Simplify the final answer as much as possible. Solve the equation $\sin 4 \theta+\sin 2 \theta=0$ on the interval $0 \leq \theta<2 \pi$ . Answers must be exact.

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Problem 1253

Exercice 02 : La résultante de deux forces $\vec{F}_{1}$ et $\vec{F}_{2}$ est égale à 50 N et fait un angle de $30^{\circ}$ avec la force $F_{1}=15 \mathrm{~N}$ . Trouver le module de la force $\vec{F}_{2}$ et l'angle entre les deux forces.

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Problem 1254

\begin{align} \text{Match the functions with their graphs.} \\
1. & \quad f(x) = \cos(x) \\
2. & \quad f(x) = \sin(x) \\
3. & \quad f(x) = \tan(x) \\
4. & \quad f(x) = \arcsin(x) \\
5. & \quad f(x) = \arccos(x) \\
6. & \quad f(x) = \arctan(x) \\ \text{Graphs:} \\ A & \\ B & \\ C & \\ D & \\ E & \\ F & \\ \end{align}

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Problem 1255

The law of sines The law of sines says that if $a, b$ , and $c$ are the sides opposite the angles $A, B$ , and $C$ in a triangle, then $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} .$
Use the accompanying figures and the identity $\sin (\pi-\theta)=$ $\sin \theta$ , if required, to derive the law.

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Problem 1256

Convert the polar equation to a rectangular equation. (10.7, Example 5) $\theta=\frac{5 \pi}{6}$

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Problem 1257

A. -8 B. -2 C. -14 D. -4 E.None Q.6) $\cos ^{2}\left(\tan ^{-1} x\right)=$ A. $x^{2}+1$ B. $\frac{x^{2}}{x^{2}+1}$ C. $\frac{1}{x^{2}+1}$ D. $\frac{1}{x^{2}}$ E.None

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Problem 1258

NATIONAL 5 MATHS 2014 126
10. The graph of $y=a \sin (x+b)^{\circ}, 0 \leq x \leq 360$ , is shown below.
Write down the values of $a$ and $b$ .

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Problem 1259

NATIONAL 5 MATHS 2014 126
10. The graph of $y=a \sin (x+b)^{\circ}, 0 \leq x \leq 360$ , is shown below.
Write down the values of $a$ and $b$ .

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Problem 1260

For the given function, use your intuition or additional research, if necessary, to complete parts (a) through (c) below. (angle of football, horizontal distance traveled by football) a. Describe an appropriate domain and range for the function. b. Make a rough sketch of a graph of the function. c. Briefly discuss the validity of the graph as a model of the true function. a. Choose the appropriate domain for the function below. A. The domain is the horizontal distances traveled by the football thrown at certain angles or 0 ft to 200 mi . B. The domain is all angles a football could be thrown at or $0^{\circ}$ to $360^{\circ}$ . C. The domain is all angles a football could be thrown at or $0^{\circ}$ to $90^{\circ}$ . D. The domain is the horizontal distances traveled by the football thrown at certain angles or 0 ft to 200 ft .

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Problem 1261

A. $\frac{\pi}{2}$ B. $\frac{3 \pi}{2}$ C. $\pi$ D.undefined Q.27) The value of $4 \sin ^{-1}\left(\frac{\sqrt{5}}{19}\right)+4 \cos ^{-1}\left(\frac{\sqrt{5}}{19}\right)=$ A. $\pi$ B. $\frac{1}{2} \pi$ C. $\frac{1}{4} \pi$ D. $2 \pi$

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Problem 1262

Q.42) The range of the function $f(x)=2 \sec x$ equals A. R B. $R-[-2,2]$ C. R-(-2,2) D. $(-2,2)$

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Problem 1263

Which of the following values for $\theta$ is a counterexample to the claim that $\cos (\pi-\theta)=\cos (\theta)$ is an identity? a) $\frac{3 \pi}{2}$ b) $\pi$ c) $\frac{\pi}{2}$ d) $\cos (\pi-\theta)=\cos (\theta)$ is an identity.

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Problem 1264

Trigonomet'y Homework
A flagpole $[G H]$ , shown in the diagram, is vertical and the ground is inclined at an angle of $5^{\circ}$ to the horizontal between $E$ and $G$ . The angles of elevation from $E$ and $F$ to the top of the pole are $35^{\circ}$ and $52^{\circ}$ respectively. The distance from $E$ to $F$ along the incline is 6 m . Find how far $F$ is from the base of the pole $(G)$ along the incline. Give your answer correct to two decimal places.

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Problem 1265

Which of the following is a simplification of $\sin 60^{\circ} \cos 15^{\circ}-\sin 15^{\circ} \cos 60^{\circ}$ a) $\frac{\sqrt{2}}{2}$ b) $\frac{1}{2}$ C) $\frac{\sqrt{3}}{2}$ d) $\frac{\sqrt{3}+1}{2 \sqrt{2}}$

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Problem 1266

Simplify $\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}$ . a) 0.966 b) $\frac{1}{2}$ C) $\frac{\sqrt{3}}{2}$ d) $\frac{\sqrt{2}}{2}$

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Problem 1267

Which of the following properly expresses $\cos 120^{\circ}$ with a compound angle formula? a) $\cos 90^{\circ} \cos 30^{\circ}+\sin 90^{\circ} \sin 30^{\circ}$ b) $\sin 90^{\circ} \cos 30^{\circ}-\sin 30^{\circ} \cos 90^{\circ}$ C) $\sin 90^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 90^{\circ}$ d) $\cos 90^{\circ} \cos 30^{\circ}-\sin 90^{\circ} \sin 30^{\circ}$

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Problem 1268

A 100 -pound weight is to be dragged up a $20^{\circ}$ ramp. We want to know how hard to pull on the cable to move the weight up the ramp (if friction is ignored). That is, we need to know the magnitude of the component of the weight vector in the direction opposite the cable. How hard do we need to pull on the cable?

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Problem 1269

a) Show that the cosine rule shown below can be rearranged to give $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ b) What is the size of angle $\theta$ in the triangle below? Give your answer to the nearest degree.
Not drawn accurately Zoom

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Problem 1270

5. Forces of 15 N and 11 N act a point at $125^{\circ}$ to each other. Find the magnitude of the resultant.

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Problem 1271

a) Determine $\csc \theta, \sec \theta$ and $\cot \theta$ b) Calculate $\theta$ to the nearest degrees.

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Problem 1272

A ship is sailing east. At one point, the bearing of a submerged rock is $46^{\circ} 20^{\prime}$ . After the ship has sailed 13.7 mi , the bearing of the rock has become $307^{\circ} 40^{\prime}$ . Find the distance of the ship from the rock at the latter point.
The distance is approximately $\square$ mi. (Do not round until the final answer. Then round to the nearest tenth as needed.)

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Problem 1273

Find an equation for the graph shown to the right.
Type the equation in the form $y=A \sin (\omega x)$ or $y=A \cos (\omega x)$ . $\mathrm{y}=\square$ (Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

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Problem 1274

Find an equation for the graph shown to the right.
Type the equation in the form $y=A \sin (\omega x)$ or $y=A \cos (\omega x)$ . $y=\square$ (Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

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Problem 1275

Find an equation for the graph shown to the right.
Type the equation in the form $y=A \sin (\omega x)$ or $y=A \cos (\omega x)$ . $y=\square$ (Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

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Problem 1276

Question 18, *7.3.61
Solve the equation on the interval $0 \leq \theta<2 \pi$ . $\sqrt{2} \cos ^{2} \theta-\cos \theta=0$
Select the correct choice and fill in any answer boxes in your choice below. A. The solution set is $\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed. Type your answer in radians Use any numbers in the expression. Use a comma to separate answers as needed.) B. There is no solution.

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Problem 1277

Problem 2. (1 point) Find all solutions $x$ to the equation $6 \cos \left(\frac{\pi}{4} x\right)=5.5$ in the interval $[0,6]$ (if there is thore than one solution, separate them with commas). $x=\square .$

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Problem 1278

Aab-5-wDemo-STA $\times \mid$ Dal Lab 16 - MATH-1018-A01 - Pre-C $\times \mid$ bat Assignment 11 - MATH-1018-A01 $\times \mid \square$ My Student Aid Assignment 11 | Knewton knewton.com/learn/course/9e2ba2d3-121d-42f9-bfd7-dd6829a2a6b3/assignment/6db1e2bb-00c9-41da-b33c-1012b9fe79c1 MATH-018-A01 - PRECCNLOULUS IN PRACTICE MSTERY $\leftarrow$ Assignment 11 URRENT OBJECTIVE SWITCH Use sinusoidal functions to solve real-world applications
Question The equation $T=0.6 \sin \left(\frac{\pi}{28}(t-1)\right)+101.0$ describes the body temperature of a pig in Fahrenheit, where $t$ is time in hours. What is the temperature of the pig to the nearest tenth of a degree when $t=3$ ? Do not include the units in your answer. NOTE: The angle is in radians.
Provide your answer below: \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline ( & & \multicolumn{5}{|c|}{ $\checkmark$ } & (1) \\ \hline 7 & 8 & 9 & $\div$ & $x$ & $y$ & $x^{2}$ & $\sqrt{ }$ \\ \hline 4 & 5 & 6 & × & $\frac{x}{\square}$ & $x \frac{\square}{\square}$ & $x \square$ & $x_{\square}$ \\ \hline 1 & 2 & 3 & - & < & $>$ & $\pm$ & \ \\ \hline 0 & - & , & + & \% & 。 & : & ( \square \\ \hline 4 & - & 6 & = & | | | & \pi$ & ） & \\ \hline \end{tabular} Search

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Problem 1279

Question The equation $p=7000 \cos \left(\frac{\pi}{10} t\right)+45000$ describes the number of deer in a forest where $t$ is the number of years after 1972. What was the population in the year 1978 to the nearest whole number? NOTE: The angle is in radians.
Provide your answer below: $p=\square]^{\text {deer }}$

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Problem 1280

Question The equation $d=5 \sin (24 t)$ measures the displacement of a swinging pendulum in simple harmonic motion. $t$ is measured in seconds and $d$ is measured in centimeters. What is the displacement, to the nearest centimeter, when $t=4$ ? NOTE: The angle is in radians.
Provide your answer below: $d=$ $\square$ Icm

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Problem 1281

Question Given that $\cot (x)=-\frac{5}{2}$ , what is $\cot (-x)$ ?
Provide your answer below:

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Problem 1282

8. Determine all possible values for $x$ when $0 \leq x \leq 2 \pi$ for each of the following. Give answers in simplest exact form. [A6] a. $\sin \frac{8 \pi}{23}=\sin x, \quad x \neq \frac{8 \pi}{23}$

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Problem 1283

Here is a little more review concerning trig functions. Using the formula for $\sin ($ and $\cos \theta$ of the sum of two angles. $\begin{array}{l} 3 \cos (5 x-2)=3 \cos (2) \\ 3 \sin (2 x-2)=-3 \sin (2)-\square \cos (5 x)-\square \sin (5 x) \\ \cos (2 x)+3 \cos (2) \end{array}$
Now reverse this formula and given the expanded version find the version with just one term. This involves solving a pair of equations -in order to get $A \cos (x)+B \sin (x)=R \sin (x+b)=R \sin (b) \cos (x)+R \cos (b) \sin (x)$ what values must you choose for $R$ and $b$ ? (Match coefficients.) By convention we'll assume that the amplitude (the first coefficient on the left hand side) is positive. $\begin{array}{l} \cos (5 x+\square)=4 \cos (5 x)+-2 \sin (5 x) \\ \sin (2 x+\arctan (3) \quad)=6 \cos (2 x)+2 \sin (2 x) \end{array}$
The upshot of this exercise is that we can always rewrite the sum of multiples of $\sin ()$ and $\cos 0$ as a single $\sin 0$ function with a given amplitude and phase shift. We could also write it as a single $\cos ($ ), but it would have a different phase in that case. We'll use this many times in interpreting results.

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Problem 1284

Cled inly seiection
9 Multiple Choice 1 point
An airplane leaves the runway climbing at an angle of $\theta=\frac{\pi}{10}$ and with a speed of 275 feet per second. Find the altitude of the plane after 2 minutes. Round your answer to the nearest tenth. 10,197.6 feet 784.6 feet 15,692.4 feet 3386.1 feet
Clear my selection

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Problem 1285

Analyze the function $L = 2\cos(x) + 1$ and describe its general behavior.

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Problem 1286

5. Solve each equation in the specified domain. a) $3 \cos \theta-1=4 \cos \theta, 0 \leq \theta<2 \pi$ b) $\sqrt{3} \tan \theta+1=0,-\pi \leq \theta \leq 2 \pi$ c) $\sqrt{2} \sin x-1=0,-360^{\circ}<x \leq 360^{\circ}$ d) $3 \sin x-5=5 \sin x-4$ , $-360^{\circ} \leq x<180^{\circ}$ e) $3 \cot x+1=2+4 \cot x$ , $-180^{\circ}<x<360^{\circ}$

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Problem 1287

SOH CAH TOA $\sin \theta=\frac{o p p}{h y p} \quad \cos \theta=\frac{a d j}{h y p} \quad \tan \theta=\frac{o p p}{a d j}$
1. Find the missing side. Round to the nearest tenth. Show all your work!

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Problem 1288

A 56.0 cm long wrench is used to remove a lug nut from a car tire. If the torque needed to remove it is 107 Nm and you apply a force at the end of the wrench at an angle of $75.0^{\circ}$ , how much force should you apply? 738 N 198 N 111 N 191 N
Question 3

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Problem 1289

ng and reasoning. MEDIUM O Write a sinusoidal equation that has the following characteristics: The midline is at y = 5 The length of one period is 60° y= 4 sin ②x-60°).

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Problem 1290

\begin{align} \text{Pour l'équation } y = -3 \cos \left(2\left(x + 45^{\circ}\right)\right) + 2, \\ \text{trouver les valeurs de BERT qui permettent de tracer le graphique.} \\ \text{Donnez les nombres exacts pour chaque B, E, R, T.} \end{align}

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Problem 1291

Question Given the side length $c=15$ and the angle $A=18^{\circ}$ on the triangle below, find the lengths of $a$ and $b$ and the measure of angle $B$ . Do not round during your calculations, but round your final answers to one decimal place.
Provide your answer below: $a=\square, b=\square=\square .$

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Problem 1292

Solve the equation. Give a general formula for all the solutions. List six solutions. $\sin \theta=-\frac{1}{2}$
Identify the general formula for all the solutions to $\sin \theta=-\frac{1}{2}$ based on the smaller angle. $\theta=$ $\square$ k is an integer (Simplify your answer. Use angle measureaz greater than or equal to 0 and less than $2 \pi$ . Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression. Type an expression using $k$ as the variable.)

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Problem 1293

Graph $r=2 \cos \theta$ by filling out the following table and plotting the points. Then connect the points using a smooth curve to get the graph. \begin{tabular}{|l|l|} \hline $\theta$ & \\ \hline 0 & $2 \cos 6=0,(0,0)$ \\ \hline $\frac{\pi}{6}$ & 2006 \\ \hline $\frac{\pi}{4}$ & \\ \hline $\frac{\pi}{3}$ & \\ \hline $\frac{\pi}{2}$ & \\ \hline $\frac{2 \pi}{3}$ & \\ \hline $\frac{3 \pi}{4}$ & \\ \hline $\frac{5 \pi}{6}$ & \\ \hline $\pi$ \\ \hline \end{tabular}

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Problem 1294

If $\theta=\frac{\pi}{9}$ radians, what is the value of $\theta$ in degrees?

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Problem 1295

OSHA safety regulations require that the base of a ladder be placed 1 ft from the wall for every 4 ft of ladder length. To the nearest tenth of a degree, find the angle that the ladder forms with the ground and the angle that it forms with the wall.

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Problem 1296

\text{Determine which days of the year the sun rose at 7am, disregarding leap years, using the formula:} \\ t = -1.79 \sin \left[\frac{2 \pi}{365}(d-79)\right] + 6.3 \\ \text{where } t \text{ is the time of sunrise in hours and } d \text{ is the day of the year.} \\ \text{Solve for } d \text{ when } t = 7. \\

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Problem 1297

Solve $2 \sin 2 x=1.6, \quad 0 \leq x \leq 2 \pi$

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Problem 1298

11. What is the maximum value of the function $y=2 \cos \frac{2 \pi}{3}(x-7)-8$ ? [8.7] A. 2 B. -6 C. -5 D. -10

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Problem 1299

20. Write an equation that represents the sine function graphed below. [8.7]

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Problem 1300

21. Determine the exact values of the six trigonometric ratios for $585^{\circ}$ . [8.1] [2 marks]

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Trigonometry

Problem 1201

6) Convert −65.2∘-65.2^{\circ}−65.2∘ to radian measure.Round your answer to the nearest hundredth of a radian.

Problem 1202

If tan⁡θ+tan⁡30∘1−tan⁡θtan⁡30∘+1=0\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=01−tanθtan30∘tanθ+tan30∘​+1=0, find tan⁡θ\tan \thetatanθ in surd form.

Problem 1203

If tan⁡θ+tan⁡30∘1−tan⁡θtan⁡30∘+1=0\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=01−tanθtan30∘tanθ+tan30∘​+1=0, find tan⁡θ\tan \thetatanθ in surd form.

Problem 1204

Solve tan⁡θ+tan⁡30∘1−tan⁡θtan⁡30∘+1=0\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=01−tanθtan30∘tanθ+tan30∘​+1=0 for tan⁡θ\tan \thetatanθ in surd form.

Problem 1205

Solve tan⁡θ+tan⁡30∘1−tan⁡θtan⁡30∘+1=0\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=01−tanθtan30∘tanθ+tan30∘​+1=0 for tan⁡θ\tan \thetatanθ in surd form.

Problem 1206

Omar needs to find cos⁡(−2π3)\cos \left(-\frac{2 \pi}{3}\right)cos(−32π​). Use the reference angle and quadrant info to solve.

Problem 1207

Problem 1208

Problem 1209

A kite is flying 86 ft off the ground, and its string is pulled taut. The angle of elevation of the kite is 46∘46^{\circ}46∘. Find the length of the string. Round your answer to the nearest tenth. □\square□ ft

Problem 1210

Problem 1211

Problem 1212

Eliminate θ\thetaθ from the following equation (i). x=tan⁡θ+cot⁡θx=\tan \theta+\cot \thetax=tanθ+cotθ and y=tan⁡θ−cot⁡θy=\tan \theta-\cot \thetay=tanθ−cotθ

Problem 1213

Example (2) (Activity)The range of the function y=4sin⁡2θy=4 \sin 2 \thetay=4sin2θ is \qquad (3) [−4,4][-4,4][−4,4] (b) {−4,4}\{-4,4\}{−4,4} (C) [−2,2][-2,2][−2,2] (d) ]−2,2]-2,2]−2,2 [ Sol.

Problem 1214

Problem 1215

Problem 1216

Q14) The value of 4sin⁡−1(519)+4cos⁡−1(519)=4 \sin ^{-1}\left(\frac{\sqrt{5}}{19}\right)+4 \cos ^{-1}\left(\frac{\sqrt{5}}{19}\right)=4sin−1(195​​)+4cos−1(195​​)= a) π\piπ b) 12π\frac{1}{2} \pi21​π c) 14π\frac{1}{4} \pi41​π d) 2π2 \pi2π

Problem 1217

Problem 1218

Calculate 1sin⁡2x\frac{1}{\sin ^{2} x}sin2x1​

Problem 1219

tan⁡xcos⁡x=0,0≤x≤2π\tan x \cos x=0,0 \leq x \leq 2 \pitanxcosx=0,0≤x≤2π

Problem 1220

Problem 1221

11. Provide a trigonometric equation. Considering only the space between x=0x=0x=0 and 2π2 \pi2π, the equation must only have solutions at x=1x=1x=1 and x=2x=2x=2. Explain your thought process and the work you did to create the equation. You may round decimal values to 3 places. [ 6 marks]

Problem 1222

Problem 1223

Problem 1224

Example. AAA sinusoidal function has an amplitude of 2 units, a period of 180∘180^{\circ}180∘ and a maximum at (0,3)(0,3)(0,3). Write a possible equation for this function.

Problem 1225

Q3: For⁡z∈C\operatorname{For} z \in \mathbb{C}Forz∈C, show that: (a) sin⁡zˉ=sin⁡z‾\sin \bar{z}=\overline{\sin z}sinzˉ=sinz; (b) cosh⁡zˉ=cosh⁡z‾\cosh \bar{z}=\overline{\cosh z}coshzˉ=coshz.

Problem 1226

The period of isSelect one:

Problem 1227

Determine the number of triangles ABCA B CABC possible with the given parts. A=42.8∘a=8.9b=10.1A=42.8^{\circ} \quad a=8.9 \quad b=10.1A=42.8∘a=8.9b=10.1How many possible solutions does this triangle have? □\square□

Problem 1228

Problem 1229

Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. cos⁡−1(cos⁡7π6)=\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=cos−1(cos67π​)= □\square□

Problem 1230

Verify that the equation is an identity. Show that csc⁡x⋅tan⁡xcos⁡x=sec⁡2x\csc x \cdot \frac{\tan x}{\cos x}=\sec ^{2} xcscx⋅cosxtanx​=sec2x

Problem 1231

Use a sum-to-product formula to find the exact value. Write your answer as a simplified fraction and rationalize the denominator, if nece sin⁡165∘−sin⁡75∘=\sin 165^{\circ}-\sin 75^{\circ}=sin165∘−sin75∘= □\square□ 延 \sqrt{\text { 延 }} 延 ​

Problem 1232

Problem 1233

Solve for xxx in the equation asec⁡x=3a \sec x = 3asecx=3.

Problem 1234

Solve for xxx if sec⁡x=3\sec x=3secx=3.

Problem 1235

Prove that 1+sin⁡xcos⁡x+cos⁡x1+sin⁡x≡2sec⁡x\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec xcosx1+sinx​+1+sinxcosx​≡2secx.

Problem 1236

Prove that 1−cos⁡2xsec⁡2x−1=1−sin⁡2x\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} xsec2x−11−cos2x​=1−sin2x.

Problem 1237

Show that sin⁡x1−cos⁡2x=cosec⁡x\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x1−cos2xsinx​=cosecx.

Problem 1238

Prove that sec⁡4θ−sec⁡2θ=tan⁡2θ+tan⁡4θ\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \thetasec4θ−sec2θ=tan2θ+tan4θ.

Problem 1239

Prove that 1+sin⁡x1−sin⁡x≡(tan⁡x+sec⁡x)2\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}1−sinx1+sinx​≡(tanx+secx)2.

Problem 1240

Prove that (1+sin⁡xcos⁡x)2+(1−sin⁡xcos⁡x)2=2sec⁡2x+2tan⁡2x\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x(cosx1+sinx​)2+(cosx1−sinx​)2=2sec2x+2tan2x.

Problem 1241

Solve 9cot⁡212x=49 \cot ^{2} \frac{1}{2} x=49cot221​x=4 for −180∘≤x≤180∘-180^{\circ} \leq x \leq 180^{\circ}−180∘≤x≤180∘.

Problem 1242

Solve 2cosec⁡(2x−1)=32 \operatorname{cosec}(2 x-1)=32cosec(2x−1)=3 for −π≤x≤π-\pi \leq x \leq \pi−π≤x≤π.

Problem 1243

Determine the other trigonometric functions for θ\thetaθ given that tan⁡θ=−16\tan \theta=-\frac{1}{6}tanθ=−61​ and sin⁡θ>0\sin \theta>0sinθ>0.

Problem 1244

Find the exact value of tan⁡(−π3)\tan \left(-\frac{\pi}{3}\right)tan(−3π​) using reference angles.

Problem 1245

Find the exact value of sin⁡(−300∘)\sin \left(-300^{\circ}\right)sin(−300∘) using reference angles.

Problem 1246

6) Convert $-65.2^{\circ}$ to radian measure.
Round your answer to the nearest hundredth of a radian.

If $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ , find $\tan \theta$ in surd form.

If $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ , find $\tan \theta$ in surd form.

Solve $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ for $\tan \theta$ in surd form.

Solve $\frac{\tan \theta+\tan 30^{\circ}}{1-\tan \theta \tan 30^{\circ}}+1=0$ for $\tan \theta$ in surd form.

Omar needs to find $\cos \left(-\frac{2 \pi}{3}\right)$ . Use the reference angle and quadrant info to solve.

A kite is flying 86 ft off the ground, and its string is pulled taut. The angle of elevation of the kite is $46^{\circ}$ . Find the length of the string. Round your answer to the nearest tenth. $\square$ ft

Eliminate $\theta$ from the following equation (i). $x=\tan \theta+\cot \theta$ and $y=\tan \theta-\cot \theta$

Example (2) (Activity)
The range of the function $y=4 \sin 2 \theta$ is $\qquad$ (3) $[-4,4]$ (b) $\{-4,4\}$ (C) $[-2,2]$ (d) $]-2,2$ [ Sol.

Q14) The value of $4 \sin ^{-1}\left(\frac{\sqrt{5}}{19}\right)+4 \cos ^{-1}\left(\frac{\sqrt{5}}{19}\right)=$ a) $\pi$ b) $\frac{1}{2} \pi$ c) $\frac{1}{4} \pi$ d) $2 \pi$

Calculate $\frac{1}{\sin ^{2} x}$

$\tan x \cos x=0,0 \leq x \leq 2 \pi$

11. Provide a trigonometric equation. Considering only the space between $x=0$ and $2 \pi$ , the equation must only have solutions at $x=1$ and $x=2$ . Explain your thought process and the work you did to create the equation. You may round decimal values to 3 places. [ 6 marks]

Example. $A$ sinusoidal function has an amplitude of 2 units, a period of $180^{\circ}$ and a maximum at $(0,3)$ . Write a possible equation for this function.

Q3: $\operatorname{For} z \in \mathbb{C}$ , show that: (a) $\sin \bar{z}=\overline{\sin z}$ ; (b) $\cosh \bar{z}=\overline{\cosh z}$ .

The period of is
Select one:

Determine the number of triangles $A B C$ possible with the given parts. $A=42.8^{\circ} \quad a=8.9 \quad b=10.1$
How many possible solutions does this triangle have? $\square$

Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=$ $\square$

Verify that the equation is an identity. Show that $\csc x \cdot \frac{\tan x}{\cos x}=\sec ^{2} x$

Use a sum-to-product formula to find the exact value. Write your answer as a simplified fraction and rationalize the denominator, if nece $\sin 165^{\circ}-\sin 75^{\circ}=$ $\square$ $\sqrt{\text { 延 }}$

Solve for $x$ in the equation $a \sec x = 3$ .

Solve for $x$ if $\sec x=3$ .

Prove that $\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv 2 \sec x$ .

Prove that $\frac{1-\cos ^{2} x}{\sec ^{2} x-1} = 1-\sin ^{2} x$ .

Show that $\frac{\sin x}{1-\cos^{2} x} = \operatorname{cosec} x$ .

Prove that $\sec ^{4} \theta - \sec ^{2} \theta = \tan ^{2} \theta + \tan ^{4} \theta$ .

Prove that $\frac{1+\sin x}{1-\sin x} \equiv(\tan x+\sec x)^{2}$ .

Prove that $\left(\frac{1+\sin x}{\cos x}\right)^{2}+\left(\frac{1-\sin x}{\cos x}\right)^{2} = 2 \sec ^{2} x+2 \tan ^{2} x$ .

Solve $9 \cot ^{2} \frac{1}{2} x=4$ for $-180^{\circ} \leq x \leq 180^{\circ}$ .

Solve $2 \operatorname{cosec}(2 x-1)=3$ for $-\pi \leq x \leq \pi$ .

Determine the other trigonometric functions for $\theta$ given that $\tan \theta=-\frac{1}{6}$ and $\sin \theta>0$ .

Find the exact value of $\tan \left(-\frac{\pi}{3}\right)$ using reference angles.

Find the exact value of $\sin \left(-300^{\circ}\right)$ using reference angles.

Find the exact value of $\cos \left(-\frac{\pi}{3}\right)$ using reference angles.

In $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

A 15 ft sliding board is at a $40^{\circ}$ angle. Find the height to the nearest tenth.

$\sin ^{-1}(1.94)=$ $\square$ $\tan ^{-1}(2.27)=$ $\square$ $\cos ^{-1}(-0.62)=$ $\square$

$\begin{array}{c}\tan ^{-1}(-1.53)= \\ \cos ^{-1}(2.26)= \\ \sin ^{-1}(0.42)=\end{array}$

Simplify the final answer as much as possible. Solve the equation $\sin 4 \theta+\sin 2 \theta=0$ on the interval $0 \leq \theta<2 \pi$ . Answers must be exact.

Exercice 02 : La résultante de deux forces $\vec{F}_{1}$ et $\vec{F}_{2}$ est égale à 50 N et fait un angle de $30^{\circ}$ avec la force $F_{1}=15 \mathrm{~N}$ . Trouver le module de la force $\vec{F}_{2}$ et l'angle entre les deux forces.

Convert the polar equation to a rectangular equation. (10.7, Example 5) $\theta=\frac{5 \pi}{6}$

A. -8 B. -2 C. -14 D. -4 E.None Q.6) $\cos ^{2}\left(\tan ^{-1} x\right)=$ A. $x^{2}+1$ B. $\frac{x^{2}}{x^{2}+1}$ C. $\frac{1}{x^{2}+1}$ D. $\frac{1}{x^{2}}$ E.None

NATIONAL 5 MATHS 2014 126
10. The graph of $y=a \sin (x+b)^{\circ}, 0 \leq x \leq 360$ , is shown below.
Write down the values of $a$ and $b$ .

NATIONAL 5 MATHS 2014 126
10. The graph of $y=a \sin (x+b)^{\circ}, 0 \leq x \leq 360$ , is shown below.
Write down the values of $a$ and $b$ .

Q.42) The range of the function $f(x)=2 \sec x$ equals A. R B. $R-[-2,2]$ C. R-(-2,2) D. $(-2,2)$

Simplify $\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}$ . a) 0.966 b) $\frac{1}{2}$ C) $\frac{\sqrt{3}}{2}$ d) $\frac{\sqrt{2}}{2}$

a) Show that the cosine rule shown below can be rearranged to give $\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c}$ b) What is the size of angle $\theta$ in the triangle below? Give your answer to the nearest degree.
Not drawn accurately Zoom

5. Forces of 15 N and 11 N act a point at $125^{\circ}$ to each other. Find the magnitude of the resultant.

a) Determine $\csc \theta, \sec \theta$ and $\cot \theta$ b) Calculate $\theta$ to the nearest degrees.