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exact value of sin 165

Understand the Problem

The question is asking for the exact value of the sine of 165 degrees. To solve this, we can use the sine addition formulas or known values of sine for associated angles.

Answer

The exact value of $\sin(165^\circ)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$.
Answer for screen readers

The exact value of $\sin(165^\circ)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$.

Steps to Solve

  1. Identify Related Angles

We can express $165^\circ$ as the sum of two known angles: $165^\circ = 120^\circ + 45^\circ$.

  1. Use Sine Addition Formula

We apply the sine addition formula:

$$ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) $$

In this case, let $a = 120^\circ$ and $b = 45^\circ$.

  1. Evaluate Sine and Cosine for Known Angles

Now we need the sine and cosine values:

  • $ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} $
  • $ \cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2} $
  • $ \sin(45^\circ) = \frac{\sqrt{2}}{2} $
  • $ \cos(45^\circ) = \frac{\sqrt{2}}{2} $
  1. Substitute Values into the Formula

Substituting the values into the sine addition formula:

$$ \sin(165^\circ) = \sin(120^\circ) \cos(45^\circ) + \cos(120^\circ) \sin(45^\circ $$

This becomes:

$$ \sin(165^\circ) = \left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \right) + \left(-\frac{1}{2} \cdot \frac{\sqrt{2}}{2} \right) $$

  1. Simplify the Expression

Now simplify the expression:

$$ \sin(165^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} $$

Combine the terms:

$$ \sin(165^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} $$

The exact value of $\sin(165^\circ)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$.

More Information

The sine of $165^\circ$ is derived using specific angle identities and the sine addition formula. Knowing the values of sine and cosine for commonly used angles like $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ can greatly simplify such problems.

Tips

  • Confusing the angle values or their corresponding sine/cosine values.
  • Forgetting the negative sign when dealing with angles in different quadrants.
  • Misapplying the formula for sine addition.
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