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 $$\theta.$$
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
$$\displaystyle \sin{\theta}=?$$
$$\displaystyle \tan{\theta}=?$$
$$\displaystyle \sec{\theta}=?$$
$$\displaystyle \csc{\theta}=?$$
$$\displaystyle \cot{\theta}=?$$

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
$$\displaystyle \cot{\theta}=\frac{1}{{ \tan{\theta}}}$$
$$\displaystyle=\frac{1}{{-\frac{7}{{24}}}}$$
$$\displaystyle=-\frac{24}{{7}}$$

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