Which of the following is equal to sin ( π/5 )? A) −cos ( π/5 ) B) −sin ( π/5 ) C) cos (3π/10 ) D) sin( 7π/10 )

Understand Trigonometric Identities: Step 1: Understand the trigonometric identities. sin(π/5) is a positive value since π/5 is in the first quadrant.Evaluate Option A: Step 2: Evaluate option A. A) −cos(π/5) - Since cos(π/5) is positive in the first quadrant, −cos(π/5) is negative. This cannot be equal to sin(π/5) which is positive.Evaluate Option B: Step 3: Evaluate option B. B) −sin(π/5) - This is simply the negative of sin(π/5), which is positive. So, this option is also incorrect.Evaluate Option C: Step 4: Evaluate option C. C) cos(3π/10) - Using the identity cos(θ) = sin(π/2 - θ), we get cos(3π/10) = sin(π/2 - 3π/10) = sin(π/5). This matches sin(π/5).Evaluate Option D: Step 5: Evaluate option D. D) sin(7π/10) - Using the identity sin(π - θ) = sin(θ), we get sin(7π/10) = sin(π - 7π/10) = sin(3π/10). This is not equal to sin(π/5).

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Which of the following is equal to $\sin \left( \frac{\pi}{5} \right)$ ? $\newline$ A) $-\cos \left( \frac{\pi}{5} \right)$ $\newline$ B) $-\sin \left( \frac{\pi}{5} \right)$ $\newline$ C) $\cos \left( \frac{3\pi}{10} \right)$ $\newline$ D) $\sin\left( \frac{7\pi}{10} \right)$

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Precalculus

Convert complex numbers between rectangular and polar form

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Q. Which of the following is equal to

\sin \left( \frac{\pi}{5} \right)

\newline

-\cos \left( \frac{\pi}{5} \right)

\newline

-\sin \left( \frac{\pi}{5} \right)

\newline

\cos \left( \frac{3\pi}{10} \right)

\newline

\sin\left( \frac{7\pi}{10} \right)

Understand Trigonometric Identities: Step $1$ : Understand the trigonometric identities. $\sin(\frac{\pi}{5})$ is a positive value since $\frac{\pi}{5}$ is in the first quadrant.
Evaluate Option A: Step $2$ : Evaluate option A. $\newline$ A) $−\cos(\frac{\pi}{5})$ - Since $\cos(\frac{\pi}{5})$ is positive in the first quadrant, $−\cos(\frac{\pi}{5})$ is negative. This cannot be equal to $\sin(\frac{\pi}{5})$ which is positive.
Evaluate Option B: Step $3$ : Evaluate option B. $\newline$ B) $-\sin(\frac{\pi}{5})$ - This is simply the negative of $\sin(\frac{\pi}{5})$ , which is positive. So, this option is also incorrect.
Evaluate Option C: Step $4$ : Evaluate option C. $\newline$ C) $\cos(\frac{3\pi}{10})$ - Using the identity $\cos(\theta) = \sin(\frac{\pi}{2} - \theta)$ , we get $\cos(\frac{3\pi}{10}) = \sin(\frac{\pi}{2} - \frac{3\pi}{10}) = \sin(\frac{\pi}{5})$ . This matches $\sin(\frac{\pi}{5})$ .
Evaluate Option D: Step $5$ : Evaluate option D. $\newline$ D) $\sin(\frac{7\pi}{10})$ - Using the identity $\sin(\pi - \theta) = \sin(\theta)$ , we get $\sin(\frac{7\pi}{10}) = \sin(\pi - \frac{7\pi}{10}) = \sin(\frac{3\pi}{10})$ . This is not equal to $\sin(\frac{\pi}{5})$ .