Identities & Equations

Problem 401

Question 19
Solve for the exact solutions in the interval [0,2π)[0, 2\pi). List your answers separated by a comma, if it has no real solutions, enter DNE.
sin(x2)=2sin(x2)\sin(\frac{x}{2}) = \sqrt{2} - \sin(\frac{x}{2})
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Problem 402

Evaluate the expression cos1(sin(4π3))\cos^{-1}\left(\sin\left(\frac{4\pi}{3}\right)\right).

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Problem 403

sinθ=33 \sin \theta = \frac{\sqrt{3}}{3} , π2<θ<π \frac{\pi}{2} < \theta < \pi (a) sin(2θ)= \sin(2\theta) = (Type an exact answer, using radicals as needed.) (b) cos(2θ) \cos(2\theta) (c) sinθ2 \sin \frac{\theta}{2} (d) cosθ2 \cos \frac{\theta}{2} Use the information given about the angle θ \theta to find the exact values of the following. Question 28, 7.6.13 Part 1 of 4 LM est. 60 min

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Problem 404

6 Se x[π2;0]x \in \left[ -\frac{\pi}{2}; 0 \right] e cos(xπ4)=cos(2x+π3)\cos\left(x - \frac{\pi}{4}\right) = \cos\left(2x + \frac{\pi}{3}\right) allora xx è uguale a: a π6-\frac{\pi}{6} b π12-\frac{\pi}{12} c π18-\frac{\pi}{18} d π36-\frac{\pi}{36}

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Problem 405

Question 3 (1 point) Factor the expression sin2θsinθ12\sin ^{2} \theta-\sin \theta-12. (sinθ3)(sinθ+4)(\sin \theta-3)(\sin \theta+4) (sinθ1)(sinθ+12)(\sin \theta-1)(\sin \theta+12) (sinθ+3)(sinθ4)(\sin \theta+3)(\sin \theta-4) (sinθ6)(sinθ+2)(\sin \theta-6)(\sin \theta+2)

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Problem 406

How many solutions does the equation cosx+12=1\frac{\cos{x} + 1}{2} = 1 have for 0x2π0 \le x \le 2\pi? 1 2 3 4

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Problem 407

Use the cosine of a sum and cosine of a difference identities to find cos(s+t)\cos (s + t) and cos(st)\cos (s - t). sins=45\sin s = -\frac{4}{5} and sint=1213\sin t = \frac{12}{13}, ss in quadrant III and tt in quadrant I cos(s+t)=\cos (s + t) = \square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

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Problem 408

conseille de refaire les roues de sorte que l'angle α[0;π3]\alpha \in [0; \frac{\pi}{3}] entre deux rayons successifs vérifie la relation : 3cos2α+sin2α=1\sqrt{3}\cos{2\alpha} + \sin{2\alpha} = 1.
Etant donné que ni Koffi, ni son soudeur n'a de connaissances en Mathématiques, il sollicite son fils Tano, votre camarade de classe pour déterminer le nombre de rayons qu'il faudra avoir sur chaque nouvelle roue. En utilisant votre connaissance mathématique au programme, Aidez votre camarade à trouver une solution au problème de son père.

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Problem 409

Simplify sin(t)sec(t)cos(t)\frac{\sin(t)}{\sec(t) - \cos(t)} to a single trig function.

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Problem 410

sec(x)cot(x)=\sec (x) \cot (x)=
A sin(x)\sin (x)
B cos(x)\cos (\mathrm{x})
C tan(x)\tan (x)
D sec(x)\sec (x) Ecsc(x)E \quad \csc (x)

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Problem 411

Convert r=4cos(θ)r=4 \cos (\theta) to a rectangular equation.

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Problem 412

cos2xcos4xsin2x+sin4x=cos2x+cos4xsin2xsin4x=\begin{array}{l} \frac{\cos 2 x-\cos 4 x}{\sin 2 x+\sin 4 x}= \\ \frac{\cos 2 x+\cos 4 x}{\sin 2 x-\sin 4 x}= \end{array} tanx\tan x \qquad cots -Cot xx sinx\sin x tanx-\tan x

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Problem 413

3
Select the correct answer.
Convert x=4x=4 to polar form. A. r=4sinθr=4 \sin \theta B. r=cosθ4r=\frac{\cos \theta}{4} C. r=4cosθr=\frac{4}{\cos \theta} D. r=4r=4 E. r=4secθr=\frac{4}{\sec \theta}

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Problem 414

1. cos(π/9)0.9397\cos (\pi / 9) \cong 0.9397. use equivalent trigonometric expression to evaluate sin(7π/18)\sin (7 \pi / 18) Show at least three lines of work for full marks. 2 marks
2. Use trigonometric identities and compound angle formulas to calculate the exact value of sin19π22\sin \frac{19 \pi}{22}, Simplify your answer and rationalize denominators. Show at least four lines of work for full marks, 4 marks

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Problem 415

7. Write the expression cot2θ+tan2θcsc2θ+2\cot ^{2} \theta+\tan ^{2} \theta-\csc ^{2} \theta+2 as a single term. [sect. 9.4] A. cos2θ-\cos ^{2} \theta B. secθ\sec \theta C. sec2θ-\sec ^{2} \theta D. sec2θ\sec ^{2} \theta

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Problem 416

Prove that 2cos(x)12cos2(x)7cos(x)+3=1cosx3\frac{2 \cos (x)-1}{2 \cos ^{2}(x)-7 \cos (x)+3}=\frac{1}{\cos x-3} is an identity.

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Problem 417

12cos540+23sin72014sin450+6sin(270)cos4π+2sin(152π)+13cos(3π)+sin92πsin72πcos(7π)+2sin(112π)2sin(32π)+cos4π4cos52π4(cos22π+sin252π)+8cos10π3[14cos(4π)]\begin{array}{l}\frac{1}{2} \cos 540^{\circ}+\frac{2}{3} \sin 720^{\circ}-\frac{1}{4} \sin 450^{\circ}+6 \sin \left(-270^{\circ}\right) \\ \cos 4 \pi+2 \sin \left(-\frac{15}{2} \pi\right)+\frac{1}{3} \cos (-3 \pi)+\sin \frac{9}{2} \pi \\ \frac{\sin \frac{7}{2} \pi-\cos (-7 \pi)+2 \sin \left(-\frac{11}{2} \pi\right)}{2 \sin \left(-\frac{3}{2} \pi\right)+\cos 4 \pi-4 \cos \frac{5}{2} \pi} \\ \frac{4\left(\cos ^{2} 2 \pi+\sin ^{2} \frac{5}{2} \pi\right)+8 \cos 10 \pi}{3[1-4 \cos (-4 \pi)]}\end{array}

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Problem 418

285 cosα=23e32π<α<2π;sinα\cos \alpha=\frac{2}{3} \mathrm{e} \frac{3}{2} \pi<\alpha<2 \pi ; \sin \alpha ? 186 197sinα=1213eb197 \sin \alpha=-\frac{12}{13} \mathrm{e} b \in III quadrante; cosα\cos \alpha ? 138sinα=53138 \sin \alpha=\frac{\sqrt{5}}{3} e bb \in II quadrante; cosα\cos \alpha ?

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Problem 419

EUREKAL 544 Scrivi un quadrato di binomio dal quale si ottiene 12sinαcosα1-2 \sin \alpha \cos \alpha.

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Problem 420

Find the exact value of tan1[tan(5π7)]\tan^{-1}\left[\tan\left(\frac{5\pi}{7}\right)\right] without a calculator.

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Problem 421

Find sin2x,cos2x\sin 2 x, \cos 2 x, and tan2x\tan 2 x if sinx=110\sin x=\frac{1}{\sqrt{10}} and xx terminates in quadrant II. sin2x=\sin 2 x= \square cos2x=\cos 2 x= \square

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Problem 422

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Watch the video and then solve the problem given below. Click here to watch the video. - The angle tt is an acute angle and sint\boldsymbol{\operatorname { s i n }} t and cost\boldsymbol{\operatorname { c o s }} t are given. Use identities to find tant,csct,sect\tan t, \csc t, \sec t, and cott\cot t. Where necessary, rationalize denominators. sint=16,cost=356\sin t=\frac{1}{6}, \cos t=\frac{\sqrt{35}}{6} tant=\boldsymbol{\operatorname { t a n }} \mathrm{t}= \square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) Incorrect: 0

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Problem 423

Find all exact solutions on the interval [0,2π)[0,2 \pi). If there is more than one answer, enter them as a comma separated list. 2cos2(t)+cos(t)=1t= help (angles). \begin{array}{l} 2 \cos ^{2}(t)+\cos (t)=1 \\ t=\square \text { help (angles). } \end{array}

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Problem 424

L'equazione cosx=117\cos x=\frac{1}{17} a ha quattro soluzioni c. ha più di 6 soluzioni b equivale all'equazione sen x=12172x=\frac{12}{17} \sqrt{2} d ha un'unica soluzione

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Problem 425

Which of the following expressions is NOT equivalent to sin75\sin 75^{\circ} ? a. sin45+sin30\sin 45^{\circ}+\sin 30^{\circ} b. sin(45+30)\sin \left(45^{\circ}+30^{\circ}\right) c. sin45cos30+sin30cos45\sin 45^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 45^{\circ} d. sin105\sin 105^{\circ}
2. Simplify the expression tan5π8tan3π81+tan5π8tan3π8\frac{\tan \frac{5 \pi}{8}-\tan \frac{3 \pi}{8}}{1+\tan \frac{5 \pi}{8} \tan \frac{3 \pi}{8}}. A. 0 o. undefined b. 1 d. -1
3. Which expression is a proper simplification of 2sinx2cosx22 \sin \frac{x}{2} \cos \frac{x}{2} ?
4. Pactor the expression 8116sin2θ81-16 \sin ^{2} \theta. a. (94sinθ)2(9-4 \sin \theta)^{2} c. (32sin2θ)4\left(3-2 \sin ^{2} \theta\right)^{4} я. sinx2\sin \frac{x}{2} b. 3(275sin2θ)3\left(27-5 \sin ^{2} \theta\right) d. (9+4sinθ)(94sinθ)(9+4 \sin \theta)(9-4 \sin \theta) b. sin(4x)\sin (4 x) c. sin(2x)\sin (2 x) d. sinx\sin x

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Problem 426

a) Решите уравнение: (36sin(x))cos(x)=62sin(x)\left(36^{\sin (x)}\right)^{\cos (x)}=6^{\sqrt{2} \sin (x)} б) Укажите корни этого уравнения, принадлежащие интервалу: [8π;9π][8 \pi ; 9 \pi]

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Problem 427

10 Se arcsen x=25πx=\frac{2}{5} \pi, quanto vale arccos xx ? (a) π5\frac{\pi}{5} [c] 310π\frac{3}{10} \pi [b] 35π\frac{3}{5} \pi []. π10\frac{\pi}{10}

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Problem 428

Select the correct answer.
In right triangle ABC,AA B C, \angle A and B\angle \boldsymbol{B} are complementary angles and sinA=B9\sin A=\frac{B}{9}. What is cosB\cos B ? A. 81717\frac{8 \sqrt{17}}{17} B. 89\frac{8}{9} C. 179\frac{\sqrt{17}}{9} D. 178\frac{\sqrt{17}}{8}

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Problem 429

Solve the equation: sinA1+cosA+1+cosAsinA=2cosecA\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}=2 \operatorname{cosec} A.

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Problem 430

Using Half-Angle Formulas In Exercises 73,74,7573, \underline{74}, \underline{75}, and 76\underline{76}, use the given conditions to a. determine the quadrant in which u/2u / 2 lies, and b. find the exact values of sin(u/2),cos(u/2)\sin (u / 2), \cos (u / 2), and tan(u/2)\tan (u / 2) using the half-angle formulas.
73. tanu=43,π<u<3π2\tan u=\frac{4}{3}, \quad \pi<u<\frac{3 \pi}{2}

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Problem 431

30. Find the exact values. a) cos15\cos 15^{\circ} b) sin11π12\sin \frac{11 \pi}{12}

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Problem 432

Given that sinθ=13\sin \theta=-\frac{1}{3} and that cosθ<0\cos \theta<0, then determine the exact value of tanθ\tan \theta Select one: a. 8-\sqrt{8} b. 18\frac{1}{\sqrt{8}} c. 8\sqrt{8} d. 18-\frac{1}{\sqrt{8}}

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Problem 433

12. Risolvi l'equazione 2cos2(x)3sin(x)1=02 \cos ^{2}(x)-3 \sin (x)-1=0 nell'intervallo [0,2π][0,2 \pi]. (a) x=π4,5π4x=\frac{\pi}{4}, \frac{5 \pi}{4} (b) x=π6,5π6x=\frac{\pi}{6}, \frac{5 \pi}{6} (c) x=π3,2π3,4π3,5π3x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3} (d) Nessuna soluzione (e) x=arcsin(13),πarcsin(13)x=\arcsin \left(-\frac{1}{3}\right), \pi-\arcsin \left(-\frac{1}{3}\right)

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Problem 434

12sin2xsinx+cosx+2sinx2cosx2=cosx\frac{1-2 \sin ^{2} x}{\sin x+\cos x}+2 \sin \frac{x}{2} \cos \frac{x}{2}=\cos x

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Problem 435

b) sin2x=2sinxcosx\sin 2 x=2 \sin x \cos x c) tanx=sinxcosx\tan x=\frac{\sin x}{\cos x} d) all of these - The height of the tip of one blade of a wind turbine above the ground, h(t)h(t), can be modelled by h(t)=18cos(πt+π4)+2h(t)=18 \cos \left(\pi t+\frac{\pi}{4}\right)+2 where tt is the time passed in seconds. Whic, time interval describes a period when the bl tip is at least 30 m above the ground? a) 5.24t7.335.24 \leq t \leq 7.33 (c) 1.37t21.37 \leq t \leq 2. ) 0.42t1.080.42 \leq t \leq 1.08 d) 0.08t10.08 \leq t \leq 1.
Iify cosπ5cosπ6sinπ5sinπ6\cos \frac{\pi}{5} \cos \frac{\pi}{6}-\sin \frac{\pi}{5} \sin \frac{\pi}{6}

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Problem 436

1. Which of these is an equivalent trigonomerric ratio for sin2π5\sin \frac{2 \pi}{5} ? a) cosπ10\cos \frac{\pi}{10} c) cos9π10-\cos \frac{9 \pi}{10} b) sin3π5\sin \frac{3 \pi}{5} d) all of these α=122\alpha=\frac{12}{2} and

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Problem 437

cosxsinx=0\cos x \sin x=0

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Problem 438

2. If sin(α)=25\sin (\alpha)=\frac{2}{5} and tan(β)=2\tan (\beta)=\sqrt{2}, where π2sinsm\frac{\pi}{2} \operatorname{sinsm} and π2sβs3π2\frac{\pi}{2} s \beta s^{\frac{3 \pi}{2}}, calculate sin(α+β)+sin(αβ)\sin (\alpha+\beta)+\sin (\alpha-\beta). Show at least four lines of work for fult marks. (4 Marks])
3. Determine the exact value of Cos(2x),CSc(x)=1715\operatorname{Cos}(2 x), \operatorname{CSc}(x)=\frac{-17}{15} and 3π25×2π\frac{3 \pi}{2} 5 \times \leq 2 \pi. Show at least three lines of work for full merts. (4 Marks)

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Problem 439

sin2x+sin2y=2sin(x+y)cos(xy)\sin 2x + \sin 2y = 2 \sin (x+y) \cos (x-y) Prove this using only the double angle formula, without using the sum-to-product identities.

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Problem 440

f) cos4θsin4θ=12sin2θ\cos ^{4} \theta-\sin ^{4} \theta=1-2 \sin ^{2} \theta

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Problem 441

SITUATION COMPLEXE Koffi est un grand cultivateur de la ville d'Abengourou. Il utilise une charrette pour ramener ses récoltes à la maison. La charrette comporte deux roues dotées chacune de rayons fait en barres de fer par un soudeur comme l'indique la figure ci -- contre. Il voudrait que cette charrette puisse supporter des charges beaucoup plus importantes. Sur ce, son ami Kouman, professeur de MATHS au lycée Moderne d'Abengourou lui
conseille de refaire les roues de sorte que l'angle α\alpha \in ]0; π3\frac{\pi}{3} [ entre deux rayons successifs vérifie la relation : 3cos2α+sin2α=1\sqrt{3}\cos{2\alpha} + \sin{2\alpha} = 1.
Etant donné que ni Koffi, ni son soudeur n'a de connaissances en Mathématiques, il sollicite son fils Tano, votre camarade de classe pour déterminer le nombre de rayons qu'il faudra avoir sur chaque nouvelle roue. En utilisant votre connaissance mathématique au programme, Aidez votre camarade à trouver une solution au problème de son père.

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Problem 442

Which of the following is equivalent to cos(2θ)sin2θ\frac{\cos (2 \theta)}{\sin ^{2} \theta} for all values of θ\theta for which cos(2θ)sin2θ\frac{\cos (2 \theta)}{\sin ^{2} \theta} is defined?
Select the correct answer below: secθ2sinθtanθ\sec \theta-2 \sin \theta \tan \theta 2cot2θcsc2θ2 \cot ^{2} \theta-\csc ^{2} \theta cot2θ2cos2θ\cot ^{2} \theta-2 \cos ^{2} \theta 2sinθ2 \sin \theta sinθcosθtanθ\sin \theta \cos \theta-\tan \theta

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Problem 443

2)
If sinx=45\sin x=\frac{4}{5} and siny=1213,0<x<π2,3π2<y<2π\sin y=-\frac{12}{13}, 0<x<\frac{\pi}{2}, \frac{3 \pi}{2}<y<2 \pi, evaluate can (x+y)(x+y) [4 marks]

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Problem 444

2. Determine all values of xx on the interval 0x3600^{\circ} \leq x \leq 360^{\circ} that satisfy cos(2x)=0\cos (2 x)=0.

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Problem 445

6. Solve the equation 5cos(2α)+3=05 \cos (2 \alpha)+3=0 for all values of α\alpha on the interval 0α3600^{\circ} \leq \alpha \leq 360^{\circ}. Round all answers to the nearest tenth of a degree.

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Problem 446

3. Provet sin4±sin2πsin2x=cos3xcosx\frac{\sin 4 \pm-\sin 2 \pi}{\sin 2 x}=\frac{\cos 3 x}{\cos x}

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Problem 447

Evaluate sin2(28)+cos2(28)\sin^{2}(28^{\circ}) + \cos^{2}(28^{\circ}) and simplify your answer to find the value.

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Problem 448

Find the exact value of cosπ8\cos \frac{\pi}{8}.

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Problem 449

15
Type the correct answer in the box. If necessary, use / for the fraction bar and reduce the fraction.
Complete the statement.
If cosθ=35\cos \theta=\frac{3}{5} and θ\theta is in quadrant IV, sin2θ=\sin 2 \theta= \square

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Problem 450

Verify that the equation is an identity. cscαcotα=secα\frac{\csc \alpha}{\cot \alpha}=\sec \alpha
To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step. cscαcotα=\frac{\csc \alpha}{\cot \alpha}=\frac{\square}{\square}
What transforniation is made in the numerator? \square What transformation is made in the denominator? \square

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Problem 451

Find all solutions over the interval [0,2π][0,2 \pi] given the equation 2cos2θ+3cosθ=12 \cos ^{2} \theta+3 \cos \theta=-1

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Problem 452

Accelerated Pre-Calculus Seat \# \qquad Date \qquad 4.11a - Homework Educatior Binder S
Part I: New Material - Solving Quadratic Trigonometric Equations A. Directions: Find all solutions to each equation over the interval [0,2π][0,2 \pi]. Show all wor your final answer.
1. 2cos(x)=12 \cos (x)=1
2. ) 2sin2x+3sinx+1=02 \sin ^{2} x+3 \sin x+1=0

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