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Find the value of \theta in degrees (0 ^o \theta 90^o ) and radians (0 \theta \pi /2) ....

Question:

Find the value of {eq}\theta {/eq} in degrees {eq}(0 ^o < \theta < 90^o ) {/eq} and radians {eq}(0 < \theta < \pi /2) {/eq}.

{eq}\sec \theta = 1.154700538 {/eq}

Trigonometric Values


There are 6 trigonometric functions corresponding to a given angle, {eq}\displaystyle \theta {/eq}

{eq}\displaystyle \sin \theta, \cos \theta, \tan \theta, \cot \theta, \sec \theta \text{ and } \csc \theta. {/eq}

Knowing one trigonometric value and the quadrant where the angle is located, we can uniquely determine the rest of the values.

For this, we may need to use some of the following trigonometric identities.

{eq}\displaystyle \sin^2 \theta+\cos^2 \theta=1\\ \displaystyle \sec \theta=\frac{1}{\cos \theta }, \displaystyle\csc \theta=\frac{1 }{\sin \theta}\\ \displaystyle \tan \theta=\frac{\sin \theta}{\cos \theta }, \displaystyle\cot \theta=\frac{\cos \theta }{\sin \theta}. {/eq}

Answer and Explanation:


Knowing the secant value of an angle in the first quadrant, {eq}\displaystyle \sec \theta= 1.154700538, 0\leq \theta\leq \frac{\pi}{2}, {/eq}

we will find the angle by using the definition of secant and inverse cosine.

{eq}\displaystyle \begin{align} \sec \theta&=\frac{1}{\cos \theta}=1.154700538\\ \implies \cos \theta&=\frac{1}{1.154700538}\\ \implies \text{ (for } \theta \text{ in the first quadrant) }\theta&=\cos^{-1}\left( \frac{1}{1.154700538}\right)=30^0\implies \boxed{\theta=30^o \text{ or }\theta =\frac{\pi}{6}\ \rm rad}. \end{align} {/eq}

To convert degrees in radians, we use the following conversion {eq}\displaystyle 1^o=\frac{\pi}{180}\ \rm radians. {/eq}


Learn more about this topic:

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Trigonometric Identities: Definition & Uses

from Honors Precalculus Textbook

Chapter 23 / Lesson 1
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