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Chapter 4: Problem 16

Suppose \(0<\theta<\frac{\pi}{2}\) and \(\tan \theta=\frac{2}{3}\). Evaluate: (a) \(\cos \theta\) (b) \(\sin \theta\)

Short Answer

Expert verified
(a) \(\cos \theta = \frac{3}{\sqrt{13}}\) (b) \(\sin \theta = \frac{2}{\sqrt{13}}\)

Step by step solution

01

Convert the given tangent value to a right triangle representation

Since we know that \(\tan \theta = \frac{2}{3}\) and the tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle, we can represent the opposite side as '2' and the adjacent side as '3' in a right triangle for the angle \(\theta\).
02

Find the hypotenuse of the right triangle using the Pythagorean theorem

The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Let the hypotenuse be represented by 'h'. Then, we can write the equation: \[h^2 = 2^2 + 3^2\]
03

Solve for the hypotenuse 'h'

Plug in the values of the opposite and adjacent sides and solve for 'h': \[h^2 = 4 + 9 = 13\] Taking the square root of both sides, we have: \[h = \sqrt{13}\]
04

Evaluate \(\cos \theta\)

The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In our case, the adjacent side is 3 and the hypotenuse is \(\sqrt{13}\). Therefore, \[\cos \theta = \frac{3}{\sqrt{13}}\]
05

Evaluate \(\sin \theta\)

The sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In our case, the opposite side is 2 and the hypotenuse is \(\sqrt{13}\). Therefore, \[\sin \theta = \frac{2}{\sqrt{13}}\] Now, we have the required values: (a) \(\cos \theta = \frac{3}{\sqrt{13}}\) (b) \(\sin \theta = \frac{2}{\sqrt{13}}\)

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Most popular questions from this chapter

In doing several of the exercises in this section, you should have noticed a relationship between \(\cos u\) and \(\sin v,\) along with a relationship between \(\sin u\) and \(\cos v\). What are these relationships? Explain why they hold.

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sin \frac{13 \pi}{12}\)

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sin \frac{25 \pi}{12}\)

Find the smallest positive number \(x\) such that $$ (\tan x)\left(1+2 \tan \left(\frac{\pi}{2}-x\right)\right)=2-\sqrt{3} $$

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\cos (-u)\)
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