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Chapter 5: Problem 7

Use the given values to find the values (if possible) of all six trigonometric functions. $$\sin x=\frac{1}{2}, \cos x=\frac{\sqrt{3}}{2}$$

Short Answer

Expert verified
\(\sin x = \frac{1}{2}\), \(\cos x = \frac{\sqrt{3}}{2}\), \(\tan x = \frac{\sqrt{3}}{3}\), \(\csc x = 2\), \(\sec x = \frac{2\sqrt{3}}{3}\), \(\cot x = \sqrt{3}\)

Step by step solution

01

Compute \(\tan x\)

The tangent of x, \(\tan x\), can be found using the formula \(\tan x = \frac{\sin x}{\cos x}\). Substituting the given values gives \(\tan x = \frac{1/2}{\sqrt{3}/2} = \frac{\sqrt{3}}{3}\)
02

Compute the reciprocals

The other three functions are reciprocals of \(\sin x\), \(\cos x\), and \(\tan x\). Hence, \(\csc x = \frac{1}{\sin x} = 2\), \(\sec x = \frac{1}{\cos x} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\), and \(\cot x = \frac{1}{\tan x} = \sqrt{3}\). Be attentive here as division by zero is not defined.
03

Round-up the Results

Given that \(\sin x = \frac{1}{2}\), \(\cos x = \frac{\sqrt{3}}{2}\), \(\tan x = \frac{\sqrt{3}}{3}\), \(\csc x = 2\), \(\sec x = \frac{2\sqrt{3}}{3}\) and \(\cot x = \sqrt{3}\), we have found the values of all six trigonometric functions.

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