Math / Trigonometry

QuestionWhich of the following expressions is NOT equivalent to $\sin 75^{\circ}$ ? a. $\sin 45^{\circ}+\sin 30^{\circ}$ b. $\sin \left(45^{\circ}+30^{\circ}\right)$ c. $\sin 45^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 45^{\circ}$ d. $\sin 105^{\circ}$
2. Simplify the expression $\frac{\tan \frac{5 \pi}{8}-\tan \frac{3 \pi}{8}}{1+\tan \frac{5 \pi}{8} \tan \frac{3 \pi}{8}}$ . A. 0 o. undefined b. 1 d. -1
3. Which expression is a proper simplification of $2 \sin \frac{x}{2} \cos \frac{x}{2}$ ?
4. Pactor the expression $81-16 \sin ^{2} \theta$ . a. $(9-4 \sin \theta)^{2}$ c. $\left(3-2 \sin ^{2} \theta\right)^{4}$ я. $\sin \frac{x}{2}$ b. $3\left(27-5 \sin ^{2} \theta\right)$ d. $(9+4 \sin \theta)(9-4 \sin \theta)$ b. $\sin (4 x)$ c. $\sin (2 x)$ d. $\sin x$

Studdy Solution

STEP 1

What is this asking? We're figuring out which expression doesn't equal $\sin(75^\circ)$ and then simplifying some trig expressions! Watch out! Remember your angle addition formulas and don't mix up sines and cosines!

STEP 2

1. Not Equivalent Expression
2. Tangent Simplification
3. Sine and Cosine Simplification
4. Factoring Trigonometric Expression

STEP 3

We're given $\sin(75^\circ)$ and some other expressions.
Let's check each one to see if it equals $\sin(75^\circ)$ .

STEP 4

Option (a) says $\sin(45^\circ) + \sin(30^\circ)$ .
This isn't the same as $\sin(45^\circ + 30^\circ)$ .
Remember, you can't just add angles inside the sine function!
So, option (a) is likely our culprit.

STEP 5

Option (b) is $\sin(45^\circ + 30^\circ)$ , which is $\sin(75^\circ)$ .
So, (b) is equivalent.

STEP 6

Option (c) is $\sin(45^\circ)\cos(30^\circ) + \sin(30^\circ)\cos(45^\circ)$ .
This is the angle addition formula for sine!
It's the same as $\sin(45^\circ + 30^\circ) = \sin(75^\circ)$ .

STEP 7

Option (d) is $\sin(105^\circ)$ .
Since $75^\circ$ and $105^\circ$ add up to $180^\circ$ , their sines are equal.
So $\sin(105^\circ) = \sin(75^\circ)$ .

STEP 8

We have the expression $\frac{\tan\frac{5\pi}{8} - \tan\frac{3\pi}{8}}{1 + \tan\frac{5\pi}{8}\tan\frac{3\pi}{8}}.$ This looks just like the tangent subtraction formula!

STEP 9

The tangent subtraction formula is $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}.$

STEP 10

In our case, $a = \frac{5\pi}{8}$ and $b = \frac{3\pi}{8}$ .
So, the expression simplifies to $\tan\left(\frac{5\pi}{8} - \frac{3\pi}{8}\right) = \tan\left(\frac{2\pi}{8}\right) = \tan\left(\frac{\pi}{4}\right).$

STEP 11

Since $\tan(\frac{\pi}{4}) = 1$ , our final simplified expression is $1$ .

STEP 12

We're asked to simplify $2\sin\frac{x}{2}\cos\frac{x}{2}$ .
This looks a lot like the double angle formula for sine!

STEP 13

The double angle formula is $\sin(2u) = 2\sin(u)\cos(u)$ .

STEP 14

In our case, $u = \frac{x}{2}$ .
So, $2\sin\frac{x}{2}\cos\frac{x}{2} = \sin\left(2 \cdot \frac{x}{2}\right) = \sin(x)$ .

STEP 15

We want to factor $81 - 16\sin^2\theta$ .
This looks like a difference of squares!

STEP 16

Remember, the difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$ .

STEP 17

We can rewrite our expression as $9^2 - (4\sin\theta)^2$ .

STEP 18

Using the difference of squares formula, we get $(9 + 4\sin\theta)(9 - 4\sin\theta)$ .

STEP 19

1. The expression that is NOT equivalent is (a).
2. The simplified expression is 1.
3. The simplified expression is $\sin(x)$ .
4. The factored expression is $(9 + 4\sin\theta)(9 - 4\sin\theta)$ .

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Studdy Solution

STEP 1

What is this asking? We're figuring out which expression *doesn't* equal sin⁡(75∘)\sin(75^\circ)sin(75∘) and then simplifying some trig expressions! Watch out! Remember your angle addition formulas and don't mix up sines and cosines!

STEP 2

1. Not Equivalent Expression2. Tangent Simplification3. Sine and Cosine Simplification4. Factoring Trigonometric Expression

STEP 3

We're given sin⁡(75∘)\sin(75^\circ)sin(75∘) and some other expressions.Let's **check each one** to see if it equals sin⁡(75∘)\sin(75^\circ)sin(75∘).

STEP 4

Option (a) says sin⁡(45∘)+sin⁡(30∘)\sin(45^\circ) + \sin(30^\circ)sin(45∘)+sin(30∘).This *isn't* the same as sin⁡(45∘+30∘)\sin(45^\circ + 30^\circ)sin(45∘+30∘).Remember, you can't just add angles inside the sine function!So, option (a) is likely our culprit.

STEP 5

Option (b) is sin⁡(45∘+30∘)\sin(45^\circ + 30^\circ)sin(45∘+30∘), which is sin⁡(75∘)\sin(75^\circ)sin(75∘).So, (b) *is* equivalent.

STEP 6

STEP 7

Option (d) is sin⁡(105∘)\sin(105^\circ)sin(105∘).Since 75∘75^\circ75∘ and 105∘105^\circ105∘ add up to 180∘180^\circ180∘, their sines are equal.So sin⁡(105∘)=sin⁡(75∘)\sin(105^\circ) = \sin(75^\circ)sin(105∘)=sin(75∘).

STEP 8

We have the expression tan⁡5π8−tan⁡3π81+tan⁡5π8tan⁡3π8. \frac{\tan\frac{5\pi}{8} - \tan\frac{3\pi}{8}}{1 + \tan\frac{5\pi}{8}\tan\frac{3\pi}{8}}. 1+tan85π​tan83π​tan85π​−tan83π​​. This looks just like the tangent subtraction formula!

STEP 9

The tangent subtraction formula is tan⁡(a−b)=tan⁡(a)−tan⁡(b)1+tan⁡(a)tan⁡(b). \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}. tan(a−b)=1+tan(a)tan(b)tan(a)−tan(b)​.

STEP 10

STEP 11

Since tan⁡(π4)=1\tan(\frac{\pi}{4}) = 1tan(4π​)=1, our **final simplified expression is** 111.

STEP 12

We're asked to simplify 2sin⁡x2cos⁡x22\sin\frac{x}{2}\cos\frac{x}{2}2sin2x​cos2x​.This looks a lot like the double angle formula for sine!

STEP 13

The double angle formula is sin⁡(2u)=2sin⁡(u)cos⁡(u)\sin(2u) = 2\sin(u)\cos(u)sin(2u)=2sin(u)cos(u).

STEP 14

In our case, u=x2u = \frac{x}{2}u=2x​.So, 2sin⁡x2cos⁡x2=sin⁡(2⋅x2)=sin⁡(x)2\sin\frac{x}{2}\cos\frac{x}{2} = \sin\left(2 \cdot \frac{x}{2}\right) = \sin(x)2sin2x​cos2x​=sin(2⋅2x​)=sin(x).

STEP 15

We want to factor 81−16sin⁡2θ81 - 16\sin^2\theta81−16sin2θ.This looks like a difference of squares!

STEP 16

Remember, the difference of squares formula is a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)a2−b2=(a+b)(a−b).

STEP 17

We can rewrite our expression as 92−(4sin⁡θ)29^2 - (4\sin\theta)^292−(4sinθ)2.

STEP 18

Using the difference of squares formula, we get (9+4sin⁡θ)(9−4sin⁡θ)(9 + 4\sin\theta)(9 - 4\sin\theta)(9+4sinθ)(9−4sinθ).

STEP 19

1. The expression that is NOT equivalent is **(a)**.2. The simplified expression is **1**.3. The simplified expression is sin⁡(x)\sin(x)sin(x).4. The factored expression is (9+4sin⁡θ)(9−4sin⁡θ)(9 + 4\sin\theta)(9 - 4\sin\theta)(9+4sinθ)(9−4sinθ).

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What is this asking? We're figuring out which expression doesn't equal $\sin(75^\circ)$ and then simplifying some trig expressions! Watch out! Remember your angle addition formulas and don't mix up sines and cosines!

1. Not Equivalent Expression
2. Tangent Simplification
3. Sine and Cosine Simplification
4. Factoring Trigonometric Expression

We're given $\sin(75^\circ)$ and some other expressions.
Let's check each one to see if it equals $\sin(75^\circ)$ .

Option (a) says $\sin(45^\circ) + \sin(30^\circ)$ .
This isn't the same as $\sin(45^\circ + 30^\circ)$ .
Remember, you can't just add angles inside the sine function!
So, option (a) is likely our culprit.

Option (b) is $\sin(45^\circ + 30^\circ)$ , which is $\sin(75^\circ)$ .
So, (b) is equivalent.

Option (d) is $\sin(105^\circ)$ .
Since $75^\circ$ and $105^\circ$ add up to $180^\circ$ , their sines are equal.
So $\sin(105^\circ) = \sin(75^\circ)$ .

We have the expression $\frac{\tan\frac{5\pi}{8} - \tan\frac{3\pi}{8}}{1 + \tan\frac{5\pi}{8}\tan\frac{3\pi}{8}}.$ This looks just like the tangent subtraction formula!

The tangent subtraction formula is $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}.$

Since $\tan(\frac{\pi}{4}) = 1$ , our final simplified expression is $1$ .

We're asked to simplify $2\sin\frac{x}{2}\cos\frac{x}{2}$ .
This looks a lot like the double angle formula for sine!

The double angle formula is $\sin(2u) = 2\sin(u)\cos(u)$ .

In our case, $u = \frac{x}{2}$ .
So, $2\sin\frac{x}{2}\cos\frac{x}{2} = \sin\left(2 \cdot \frac{x}{2}\right) = \sin(x)$ .

We want to factor $81 - 16\sin^2\theta$ .
This looks like a difference of squares!

Remember, the difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$ .

We can rewrite our expression as $9^2 - (4\sin\theta)^2$ .

Using the difference of squares formula, we get $(9 + 4\sin\theta)(9 - 4\sin\theta)$ .

1. The expression that is NOT equivalent is (a).
2. The simplified expression is 1.
3. The simplified expression is $\sin(x)$ .
4. The factored expression is $(9 + 4\sin\theta)(9 - 4\sin\theta)$ .