Find the exact value of each of the remaining trigonometric function of theta.displaystyle

Dolly Robinson 2021-01-31 Answered

Find the exact value of each of the remaining trigonometric function of \(\theta.\)
\(\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}\)
\(\displaystyle \sin{\theta}=?\)
\(\displaystyle \tan{\theta}=?\)
\(\displaystyle \sec{\theta}=?\)
\(\displaystyle \csc{\theta}=?\)
\(\displaystyle \cot{\theta}=?\)

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Expert Answer

Ayesha Gomez
Answered 2021-02-01 Author has 20460 answers

Step 1
Given:
\(\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}\)
So the angle lies in fourth quadrant. In the fourth quadrant cosine and secant are positive all others are negative.
Pythagorean identity is used for finding sine value
\(\displaystyle{{\sin}^{2}\theta}+{{\cos}^{2}\theta}={1}\)
\(\displaystyle \sin{\theta}=\pm\sqrt{{{1}-{{\cos}^{2}\theta}}}\)
\(\displaystyle=\pm\sqrt{{{1}-{\left(\frac{24}{{25}}\right)}^{2}}}\)
\(\displaystyle=\pm\sqrt{{\frac{{{25}^{2}-{24}^{2}}}{{25}^{2}}}}\)
\(\displaystyle=-\frac{7}{{25}}\)
The sine value is negative since angle is in third quadrant.
Step 2
The tan value can be found as
\(\displaystyle \tan{\theta}=\frac{{ \sin{\theta}}}{{ \cos{\theta}}}\)
\(\displaystyle=\frac{{-\frac{7}{{25}}}}{{\frac{24}{{25}}}}\)
\(\displaystyle=-\frac{7}{{24}}\)
Step 3
Secant is reciprocal of cosine.
\(\displaystyle \sec{\theta}=\frac{1}{{ \cos{\theta}}}\)
\(\displaystyle=\frac{1}{{\frac{24}{{25}}}}\)
\(\displaystyle=\frac{25}{{24}}\)
Step 4
Cosecant is the reciprocal of sine.
\(\displaystyle \csc{\theta}=\frac{1}{{ \sin{\theta}}}\)
\(\displaystyle=\frac{1}{{-\frac{7}{{25}}}}\)
\(\displaystyle=-\frac{25}{{7}}\)
Step 5
Cot is the reciprocal of tan
\(\displaystyle \cot{\theta}=\frac{1}{{ \tan{\theta}}}\)
\(\displaystyle=\frac{1}{{-\frac{7}{{24}}}}\)
\(\displaystyle=-\frac{24}{{7}}\)

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