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Chapter 5: Problem 9

Use the given values to find the values (if possible) of all six trigonometric functions. $$\cos \left(\frac{\pi}{2}-x\right)=\frac{3}{5}, \quad \cos x=\frac{4}{5}$$

Short Answer

Expert verified
The values of the six trigonometric functions are: \(\sin x = \frac{3}{5}\), \(\cos x=\frac{4}{5}\), \(\tan x = \frac{3}{4}\), \(\csc x = \frac{5}{3}\), \(\sec x = \frac{5}{4}\), \(\cot x = \frac{4}{3}\).

Step by step solution

01

Determine the value of \(\sin x\).

Since the Pythagorean identity is given by \(\sin^2x + \cos^2x = 1\), it can be rearranged to \(\sin x = \sqrt{1 - \cos^2x}\). Substitute the given value of \(\cos x\) into the equation to get \(\sin x = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}\).
02

Determine the value of \(\tan x\).

The tangent function \(\tan x\) is the ratio of the sine and cosine functions. Substitute the values of \(\sin x = \frac{3}{5}\) and \(\cos x=\frac{4}{5}\) into the definition of \(\tan x\) to get \(\tan x = \frac{sin x}{cos x} = \frac{3/5}{4/5} = \frac{3}{4}\).
03

Determine the value of \(\csc x\).

The cosecant function \(\csc x\) is the reciprocal of the sine function. Thus, \(\csc x = \frac{1}{\sin x} = \frac{5}{3}\).
04

Determine the value of \(\sec x\).

The secant function \(\sec x\) is the reciprocal of the cosine function. Thus, \(\sec x = \frac{1}{\cos x} = \frac{5}{4}\).
05

Determine the value of \(\cot x\).

The cotangent function \(\cot x\) is the reciprocal of the tangent function. Thus, \(\cot x = \frac{1}{\tan x} = \frac{4}{3}\).

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