Use a sum or difference formula to find the exact value of the trigonometric function.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Ask by Ramos Hamilton. in the United States

Jan 23,2025

Question

Use a sum or difference formula to find the exact value of the trigonometric function.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Ask by Ramos Hamilton. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the exact value of $$ \cos \frac{\pi}{12} $$, we can use the cosine difference formula. The angle $$ \frac{\pi}{12} $$ can be expressed as a difference of two angles whose cosine values we know.

We can write:
$$
\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}
$$

Now, we can apply the cosine difference formula:
$$
\cos(a - b) = \cos a \cos b + \sin a \sin b
$$
where $$ a = \frac{\pi}{3} $$ and $$ b = \frac{\pi}{4} $$.

Now we will calculate $$ \cos \frac{\pi}{3} $$, $$ \sin \frac{\pi}{3} $$, $$ \cos \frac{\pi}{4} $$, and $$ \sin \frac{\pi}{4} $$:
- $$ \cos \frac{\pi}{3} = \frac{1}{2} $$
- $$ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} $$
- $$ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} $$
- $$ \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} $$

Now substituting these values into the cosine difference formula:
$$
\cos \frac{\pi}{12} = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4}
$$
$$
= \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right)
$$
$$
= \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}
$$
$$
= \frac{\sqrt{2} + \sqrt{6}}{4}
$$

Thus, the exact value of $$ \cos \frac{\pi}{12} $$ is:
$$
\cos \frac{\pi}{12} = \frac{\sqrt{2} + \sqrt{6}}{4}
$$
$$ \cos \frac{\pi}{12} = \frac{\sqrt{2} + \sqrt{6}}{4} $$

Use a sum or difference formula to find the exact value of the trigonometric function.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

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Use a sum or difference formula to find the exact value of the trigonometric function. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

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Use a sum or difference formula to find the exact value of the trigonometric function.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)