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

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

Question

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

Answers (1)

2021-02-01
Step 1
Given:
\(\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}\)</span>
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
\(\displaystyle \cot{\theta}=\frac{1}{{ \tan{\theta}}}\)
\(\displaystyle=\frac{1}{{-\frac{7}{{24}}}}\)
\(\displaystyle=-\frac{24}{{7}}\)
0

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