What is the exact value of cos 105 degrees?
Understand the Problem
The question is asking for the exact value of the cosine function at an angle of 105 degrees. To find this, we can use the cosine sum identity or reference angles.
Answer
The exact value of $\cos(105^\circ)$ is $\frac{\sqrt{2} - \sqrt{6}}{4}$.
Answer for screen readers
The exact value of $\cos(105^\circ)$ is $\frac{\sqrt{2} - \sqrt{6}}{4}$.
Steps to Solve
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Identify the cosine angle We need to find the exact value of $\cos(105^\circ)$. We can break this angle down into smaller angles that we know the cosine values for.
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Use the cosine angle addition formula We can express $105^\circ$ as $60^\circ + 45^\circ$. Thus, we can use the cosine addition formula: $$ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $$ In this case, $a = 60^\circ$ and $b = 45^\circ$.
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Calculate cosine and sine values We need the cosine and sine values of $60^\circ$ and $45^\circ$:
- $\cos(60^\circ) = \frac{1}{2}$
- $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
- $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
- $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
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Substitute values into the cosine formula Now plug the values into the cosine addition formula: $$ \cos(105^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) $$ Substituting the known values: $$ \cos(105^\circ) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$
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Simplify the expression Simplify the expression: $$ = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} $$ Combine the terms: $$ = \frac{\sqrt{2} - \sqrt{6}}{4} $$
The exact value of $\cos(105^\circ)$ is $\frac{\sqrt{2} - \sqrt{6}}{4}$.
More Information
The cosine value of angles like 105 degrees can often be derived from known angles using identities. In this case, using the cosine of a sum formula allowed us to find a value that is not simply a standard angle on the unit circle.
Tips
- Forgetting the sign in the cosine addition formula, which can lead to incorrect values.
- Confusing the angles and their corresponding sine and cosine values.
