What is the exact value of cos(105)?
Understand the Problem
The question is asking for the exact value of the cosine of 105 degrees. To solve this, we can use the cosine angle addition formula or reference angles in the unit circle.
Answer
$$ \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4} $$
Answer for screen readers
$$ \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4} $$
Steps to Solve
- Express 105 degrees using known angles
To find $\cos(105^\circ)$, we can express it as the sum of two angles that we know the cosine values for. We can use the identity: $$ 105^\circ = 60^\circ + 45^\circ $$
- Use the cosine angle addition formula
We will apply the cosine angle addition formula: $$ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $$ Here, let $a = 60^\circ$ and $b = 45^\circ$.
- Substitute the known values
We know that:
- $\cos(60^\circ) = \frac{1}{2}$
- $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
- $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
- $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
Now substitute these values into the formula: $$ \cos(105^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) $$ This becomes: $$ \cos(105^\circ) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$
- Simplify the expression
Now let's simplify the expression: $$ \cos(105^\circ) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} $$ Combine the terms: $$ \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4} $$
$$ \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4} $$
More Information
The cosine value of $105^\circ$ can be derived from known angle values, showcasing the utility of angle addition formulas in trigonometry. This approach allows us to calculate cosine for angles that are not commonly memorized.
Tips
- A common mistake is to forget to apply the correct sign in the cosine addition formula.
- Some may confuse sine and cosine values, especially for common angles like $45^\circ$, which can lead to incorrect calculations.
