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Chapter 5: Problem 11

Use the given values to find the values (if possible) of all six trigonometric functions. $$\sec x=4, \quad \sin x>0$$

Short Answer

Expert verified
\(\cos x = \frac{1}{4}\), \(\sin x = \frac{\sqrt{15}}{4}\), \(\tan x = \sqrt{15}\), \(\csc x = \frac{4\sqrt{15}}{15}\), and \(\cot x = \frac{1}{\sqrt{15}}\).

Step by step solution

01

Find Cosine

Since \(\sec x = 4\), then \(\cos x\) is the reciprocal of 4, which is \( \cos x = \frac{1}{4}\).
02

Use Pythagorean Identity to Find Sine

Utilize the Pythagorean Identity \(\sin^2 x + \cos^2 x = 1\), substituting \(\cos x = \frac{1}{4}\) gives: \(\sin^2 x + (\frac{1}{4})^2 = 1\). Solving for \(\sin x\), we find \(\sin x = \sqrt{1 - (\frac{1}{4})^2} = \pm \frac{\sqrt{15}}{4}\). However, since \(\sin x > 0\), we must choose the positive root \(\sin x = \frac{\sqrt{15}}{4}\).
03

Find Remaining Trigonometric Functions

To find the values of the remaining functions: \(\tan x\), \(\csc x\), and \(\cot x\), use the definitions of these terms. \( \tan x = \frac{\sin x}{\cos x} = \sqrt{15}\), \(\csc x = \frac{1}{\sin x} = \frac{4\sqrt{15}}{15}\), and \(\cot x = \frac{1}{\tan x} = \frac{1}{\sqrt{15}}\).

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