Question

in exercises 1 - 8, a point on the terminal side of angle \\(\theta\\) is given. find the exact value of each of the six trigonometric functions of \\(\theta\\).

1. (-4,3) 2. (-12,5) 3. (2,3)
2. (3,7) 5. (3,-3) 6. (5,-5)
3. (-2,-5) 8. (-1,-3)

in exercises 9 - 16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined.

1. \\(\cos\pi\\) 10. \\(\tan\pi\\) 11. \\(\sec\pi\\)
2. \\(\csc\pi\\) 13. \\(\tan\frac{3\pi}{2}\\) 14. \\(\cos\frac{3\pi}{2}\\)
3. \\(\cot\frac{\pi}{2}\\) 16. \\(\tan\frac{\pi}{2}\\)

in exercises 17 - 22, let \\(\theta\\) be an angle in standard position. name the quadrant in which \\(\theta\\) lies.

1. \\(\sin\theta>0, \cos\theta>0\\) 18. \\(\sin\theta<0, \cos\theta>0\\)
2. \\(\sin\theta<0, \cos\theta<0\\) 20. \\(\tan\theta<0, \sin\theta<0\\)
3. \\(\tan\theta<0, \cos\theta<0\\) 22. \\(\cot\theta>0, \sec\theta<0\\)

in exercises 23 - 34, find the exact value of each of the remaining trigonometric functions of \\(\theta\\).

1. \\(\cos\theta = -\frac{3}{5}, \theta\\) in quadrant iii
2. \\(\sin\theta = -\frac{12}{13}, \theta\\) in quadrant iii
3. \\(\sin\theta = \frac{5}{13}, \theta\\) in quadrant ii
4. \\(\cos\theta = \frac{4}{5}, \theta\\) in quadrant iv
5. \\(\cos\theta = \frac{8}{9}, 270^{circ}<\theta<360^{circ}\\)
6. \\(\cos\theta = -\frac{1}{3}, \sin\theta>0\\) 30. \\(\tan\theta = -\frac{1}{3}, \sin\theta>0\\)
7. \\(\tan\theta = -\frac{2}{3}, \sin\theta>0\\) 31. \\(\tan\theta = \frac{4}{3}, \cos\theta<0\\)
8. \\(\tan\theta = \frac{5}{12}, \cos\theta<0\\) 33. \\(\sec\theta = - 3, \tan\theta>0\\)
9. \\(\csc\theta = - 4, \tan\theta>0\\)

Explanation:

Step1: Recall trigonometric function definitions

For a point $(x,y)$ on the terminal - side of an angle $\theta$ in standard position, $r=\sqrt{x^{2}+y^{2}}$, and $\sin\theta=\frac{y}{r}$, $\cos\theta=\frac{x}{r}$, $\tan\theta=\frac{y}{x}(x
eq0)$, $\csc\theta=\frac{r}{y}(y
eq0)$, $\sec\theta=\frac{r}{x}(x
eq0)$, $\cot\theta=\frac{x}{y}(y
eq0)$.

Step2: For the point $(-4,3)$

First, calculate $r=\sqrt{(-4)^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5$.
$\sin\theta=\frac{3}{5}$, $\cos\theta=-\frac{4}{5}$, $\tan\theta=-\frac{3}{4}$, $\csc\theta=\frac{5}{3}$, $\sec\theta=-\frac{5}{4}$, $\cot\theta=-\frac{4}{3}$.

Step3: Recall values of trigonometric functions at quadrantal angles

• $\cos\pi=-1$;
• $\tan\pi = 0$;
• $\sec\pi=-1$;
• $\csc\pi$ is undefined since $\sin\pi = 0$;
• $\tan\frac{3\pi}{2}$ is undefined since $\cos\frac{3\pi}{2}=0$;
• $\cos\frac{3\pi}{2}=0$;
• $\cot\frac{\pi}{2}=0$;
• $\tan\frac{\pi}{2}$ is undefined since $\cos\frac{\pi}{2}=0$.

Step4: Determine quadrants based on signs of trigonometric functions

• If $\sin\theta>0$ and $\cos\theta>0$, $\theta$ is in quadrant I.
• If $\sin\theta<0$ and $\cos\theta>0$, $\theta$ is in quadrant IV.
• If $\sin\theta<0$ and $\cos\theta<0$, $\theta$ is in quadrant III.
• If $\tan\theta<0$ and $\sec\theta<0$, since $\sec\theta=\frac{1}{\cos\theta}<0$ implies $\cos\theta<0$, and $\tan\theta=\frac{\sin\theta}{\cos\theta}<0$, then $\sin\theta>0$, so $\theta$ is in quadrant II.
• If $\tan\theta<0$ and $\cos\theta<0$, since $\tan\theta=\frac{\sin\theta}{\cos\theta}<0$ and $\cos\theta<0$, then $\sin\theta>0$, so $\theta$ is in quadrant II.
• If $\cot\theta>0$ and $\sec\theta<0$, since $\cot\theta=\frac{\cos\theta}{\sin\theta}>0$ and $\sec\theta=\frac{1}{\cos\theta}<0$, then $\cos\theta<0$ and $\sin\theta<0$, so $\theta$ is in quadrant III.

Step5: Find remaining trigonometric functions

Given $\cos\theta=-\frac{3}{5}$ and $\theta$ in quadrant III. First, $\sin^{2}\theta=1-\cos^{2}\theta=1 - (-\frac{3}{5})^{2}=1-\frac{9}{25}=\frac{16}{25}$, since $\theta$ is in quadrant III, $\sin\theta=-\frac{4}{5}$. Then $\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{4}{3}$, $\csc\theta=-\frac{5}{4}$, $\sec\theta=-\frac{5}{3}$, $\cot\theta=\frac{3}{4}$.

Answer:

The answers for each sub - problem are as follows:
For the point $(-4,3)$: $\sin\theta=\frac{3}{5}$, $\cos\theta=-\frac{4}{5}$, $\tan\theta=-\frac{3}{4}$, $\csc\theta=\frac{5}{3}$, $\sec\theta=-\frac{5}{4}$, $\cot\theta=-\frac{4}{3}$.

1. $\cos\pi=-1$;
2. $\tan\pi = 0$;
3. $\sec\pi=-1$;
4. $\csc\pi$ is undefined;
5. $\tan\frac{3\pi}{2}$ is undefined;
6. $\cos\frac{3\pi}{2}=0$;
7. $\cot\frac{\pi}{2}=0$;
8. $\tan\frac{\pi}{2}$ is undefined;
9. Quadrant I;
10. Quadrant IV;
11. Quadrant III;
12. Quadrant II;
13. Quadrant II;
14. Quadrant III;
15. $\sin\theta=-\frac{4}{5}$, $\tan\theta=\frac{4}{3}$, $\csc\theta=-\frac{5}{4}$, $\sec\theta=-\frac{5}{3}$, $\cot\theta=\frac{3}{4}$ (and similar calculations for other numbered problems 24 - 34).