Question

(1) Using an identity from this section, write sin 7x − sin 3x as a product.
(2) Using an identity from this section, find the exact value of sin 105◦
sin 15◦
(3) Using an identity from this section, find the exact value of sin 195◦
cos 75◦
(4) Using identities from this section, verify the identity cos t−cos 3t over sin t+sin 3t = tan t.
(5) Using an identity from this section, write sin 7x sin 3x as a sum or a difference.

Answer 1

Answer on Question # 75489 – Math – Trigonometry

Question

(1) Using an identity from this section, write $\sin 7x - \sin 3x$ as a product.

(2) Using an identity from this section, find the exact value of $\sin 105{}^\circ \sin 15{}^\circ$

(3) Using an identity from this section, find the exact value of $\sin 195{}^\circ \cos 75{}^\circ$

(4) Using identities from this section, verify the identity $\cos t - \cos 3t$ over $\sin t + \sin 3t = \tan t$.

(5) Using an identity from this section, write $\sin 7x \sin 3x$ as a sum or a difference.

Solution

1. $\sin 7x - \sin 3x = 2 \cos 5x \sin 2x$

2. $\sin 105{}^\circ \sin 15{}^\circ = \frac{1}{2} [\cos 90{}^\circ - \cos 120{}^\circ] = \frac{1}{2} x \left(-\frac{1}{2}\right) = -\frac{1}{4}$

3. $\sin 195{}^\circ \cos 75{}^\circ = \frac{1}{2} [\sin 270{}^\circ + \sin 120{}^\circ] = \frac{1}{2} [-1 + \frac{\sqrt{3}}{2}] = \frac{-2 + \sqrt{3}}{4}$

4. $\frac{\cos t - \cos 3t}{\sin t + \sin 3t} = \frac{2 \sin 2t \sin t}{2 \sin 2t \cos t} = \frac{\sin t}{\cos t} = \tan t$

5. $\sin 7x \sin 3x = \frac{1}{2} [\cos 4x - \cos 10x]$

Answer: 1. Product form of $\sin 7x - \sin 3x$ is $2 \cos 5x \sin 2x$

2. Exact value of $\sin 105{}^\circ \sin 15{}^\circ$ is $-\frac{1}{4}$

3. Exact value of $\sin 195{}^\circ \cos 75{}^\circ$ is $\frac{-2 + \sqrt{3}}{4}$

4. $\frac{\cos t - \cos 3t}{\sin t + \sin 3t} = \tan t$ (verified).

5. Expression of $\sin 7x \sin 3x$ as a difference is $\frac{1}{2} [\cos 4x - \cos 10x]$

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