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
- Identify Related Angles
We can express $165^\circ$ as the sum of two known angles: $165^\circ = 120^\circ + 45^\circ$.
- 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 $b = 45^\circ$.
- 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} $
- 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:
$$ \sin(165^\circ) = \left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \right) + \left(-\frac{1}{2} \cdot \frac{\sqrt{2}}{2} \right) $$
- 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.
