Problem 23 Find the exact values of the sin... [FREE SOLUTION]

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Chapter 6: Problem 23

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Short Answer

Expert verified

The exact values are $\sin(\frac{13 \pi}{12}) = \frac{\sqrt{2} + \sqrt{6}}{4}$, $\cos(\frac{13 \pi}{12}) = \frac{\sqrt{2} - \sqrt{6}}{4}$ and $\tan(\frac{13 \pi}{12}) = -\sqrt{3}$

Step by step solution

Split the angle

First, the angle $\frac{13 \pi}{12}$ should be expressed as a combination of common angles which we already know the sine, cosine and tangent values of. The most reasonable way would be to represent $\frac{13 \pi}{12}$ as $\frac{\pi}{4} + \frac{\pi}{3}$, because it鈥檚 easy to remember the sine, cosine and tangent of these angles, which are $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$ and $1$, $ \frac{\sqrt{3}}{2}$, respectively.

Apply sum and difference identities

With the new split, apply the sum and difference identities:The sine of sum of two angles $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$The cosine of sum of two angles $\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$ Therefore, $\sin(\frac{13 \pi}{12}) = \sin(\frac{\pi}{4} + \frac{\pi}{3}) = \sin(\frac{\pi}{4})\cos(\frac{\pi}{3}) + \cos(\frac{\pi}{4})\sin(\frac{\pi}{3})$ and $\cos(\frac{13 \pi}{12}) = \cos(\frac{\pi}{4} + \frac{\pi}{3}) = \cos(\frac{\pi}{4})\cos(\frac{\pi}{3}) - \sin(\frac{\pi}{4})\sin(\frac{\pi}{3})$

Calculation and Simplification

Substitute the known values in: $\sin(\frac{13 \pi}{12}) = \frac{\sqrt{2}}{2} \times \frac{1}{2} + \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$$\cos(\frac{13 \pi}{12}) = \frac{\sqrt{2}}{2} \times \frac{1}{2} - \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4}$Then, calculate the tangent by dividing the sine of the angle by its cosine:$\tan(\frac{13 \pi}{12}) = \frac{\sin(\frac{13 \pi}{12})}{\cos(\frac{13 \pi}{12}}) = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{\frac{\sqrt{2} - \sqrt{6}}{4}} = -\sqrt{3}$

Rationalize the denominator

For the tangential result, rationalize the denominator by multiplying and dividing by the conjugate $(\sqrt{2} + \sqrt{6})$. Then the tangent is $-\sqrt{3}$

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91影视

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Short Answer

Step by step solution

Split the angle

Apply sum and difference identities

Calculation and Simplification

Rationalize the denominator

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