Questions: Find the exact value of the expression. sin(-13π/12) The exact value of sin(-13π/12) is

Transcript text: Find the exact value of the expression. \[ \sin \left(-\frac{13 \pi}{12}\right) \] The exact value of $\sin \left(-\frac{13 \pi}{12}\right)$ is $\square$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression, all denominators.)

Solution

Solution Steps

To find the exact value of $\sin \left(-\frac{13 \pi}{12}\right)$, we can use the sum and difference identities for sine. We can express $-\frac{13 \pi}{12}$ as a sum or difference of angles whose sine and cosine values are known. One way to do this is to break it down into $-\frac{13 \pi}{12} = -\pi + \frac{\pi}{12}$. Then, we can use the sine addition formula: $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$.

Solution Approach

Express $-\frac{13 \pi}{12}$ as a sum of angles.
Use the sine addition formula to find the exact value.

Step 1: Express the Angle

We start with the angle $-\frac{13\pi}{12}$. We can express this angle as a sum of two angles: \[ -\frac{13\pi}{12} = -\pi + \frac{\pi}{12} \]

Step 2: Apply the Sine Addition Formula

Using the sine addition formula, we have: \[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \] where $a = -\pi$ and $b = \frac{\pi}{12}$.

Step 3: Calculate Sine and Cosine Values

We calculate the sine and cosine values for the angles:

$\sin(-\pi) = -1.2246467991473532 \times 10^{-16}$
$\cos(-\pi) = -1$
$\sin\left(\frac{\pi}{12}\right) \approx 0.2588$
$\cos\left(\frac{\pi}{12}\right) \approx 0.9659$

Step 4: Substitute and Simplify

Substituting these values into the sine addition formula gives: \[ \sin\left(-\frac{13\pi}{12}\right) = (-1.2246467991473532 \times 10^{-16})(0.9659) + (-1)(0.2588) \] Calculating this results in: \[ \sin\left(-\frac{13\pi}{12}\right) \approx -0.2588 \]