Problem 91
Question
You are given the value of tan \(\theta .\) Is it possible to find the value of \(\sec \theta\) without finding the measure of \(\theta ?\) Explain.
Step-by-Step Solution
Verified Answer
No, it is not possible to find the exact value of \(\sec \theta\) with just the value of \(\tan \theta\) without knowing the measure of \(\theta\). The trigonometric functions are interrelated, so the value of one does affect the range of the others. However, this relationship is reciprocal, and the exact measure of the angle \(\theta\), is necessary to determine the exact value of \(\sec \theta\).
1Step 1: Use the Pythagorean identity
The identity \(\sec^2\theta = 1 + \tan^2\theta\) relates \(\sec\theta\) and \(\tan\theta\). Given \(\tan\theta\), we get \(\sec^2\theta = 1 + \tan^2\theta\).
2Step 2: Explain the limitation
From \(\sec^2\theta\), we get \(\sec\theta = \pm\sqrt{1+\tan^2\theta}\). Without knowing which quadrant \(\theta\) is in, we cannot determine the sign of \(\sec\theta\). Therefore, we can find \(|\sec\theta|\) but not the exact value without additional information.
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