the given point lies on the terminal side of an angle $\theta$ in standard position. find the values of the six trigonometric functions of $\theta$.

1. $(3, 4)$
2. $(-6, 6)$
3. $(-4, -3)$
4. $(2, 0)$
5. $(1, -8)$
6. $(5, -3)$
7. $(-8, 15)$
8. $(-1, -2)$

find the exact value of each trigonometric function, if defined. if not defined, write undefined.

9. $\sin \frac{\pi}{2}$
10. $\tan 2\pi$
11. $\cot (-180^\circ)$
12. $\csc 270^\circ$
13. $\cos (-270^\circ)$
14. $\sec 180^\circ$
15. $\tan \pi$
16. $\sec \left(-\frac{\pi}{2}\
ight)$

sketch each angle. then find its reference angle.

17. $135^\circ$
18. $210^\circ$
19. $\frac{7\pi}{12}$
20. $\frac{11\pi}{3}$
21. $-405^\circ$
22. $-75^\circ$
23. $\frac{5\pi}{6}$
24. $\frac{13\pi}{6}$

Question

the given point lies on the terminal side of an angle $\theta$ in standard position. find the values of the six trigonometric functions of $\theta$.

1. $(3, 4)$
2. $(-6, 6)$
3. $(-4, -3)$
4. $(2, 0)$
5. $(1, -8)$
6. $(5, -3)$
7. $(-8, 15)$
8. $(-1, -2)$

find the exact value of each trigonometric function, if defined. if not defined, write undefined.

9. $\sin \frac{\pi}{2}$
10. $\tan 2\pi$
11. $\cot (-180^\circ)$
12. $\csc 270^\circ$
13. $\cos (-270^\circ)$
14. $\sec 180^\circ$
15. $\tan \pi$
16. $\sec \left(-\frac{\pi}{2}\
ight)$

sketch each angle. then find its reference angle.

17. $135^\circ$
18. $210^\circ$
19. $\frac{7\pi}{12}$
20. $\frac{11\pi}{3}$
21. $-405^\circ$
22. $-75^\circ$
23. $\frac{5\pi}{6}$
24. $\frac{13\pi}{6}$

Accepted Answer

{ "quality": "clear", "question_count": 24, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Ratios", "Pythagorean Theorem", "Radian Measure", "Degree Measure" ], "new_concepts": [ "Trigonometric Functions of General Angles", "Quadrantal Angles", "Reference Angles" ], "current_concepts": [ "Trigonometric Ratios", "Pythagorean Theorem", "Radian Measure", "Degree Measure", "Trigonometric Functions of General Angles", "Quadrantal Angles", "Reference Angles" ] } ## Find trigonometric functions for points 1 to 8 Using the *Trigonometric Ratios* and *Pythagorean Theorem* knowledge points $$ \begin{aligned} &r = \sqrt{x^2 + y^2}\ &\sin\theta = \frac{y}{r},\quad \cos\theta = \frac{x}{r},\quad \tan\theta = \frac{y}{x}\ &\csc\theta = \frac{r}{y},\quad \sec\theta = \frac{r}{x},\quad \cot\theta = \frac{x}{y} \end{aligned} $$ 1. $(3,4)$: $r = 5$. $\sin\theta = \frac{4}{5}$, $\cos\theta = \frac{3}{5}$, $\tan\theta = \frac{4}{3}$, $\csc\theta = \frac{5}{4}$, $\sec\theta = \frac{5}{3}$, $\cot\theta = \frac{3}{4}$. 2. $(-6,6)$: $r = 6\sqrt{2}$. $\sin\theta = \frac{\sqrt{2}}{2}$, $\cos\theta = -\frac{\sqrt{2}}{2}$, $\tan\theta = -1$, $\csc\theta = \sqrt{2}$, $\sec\theta = -\sqrt{2}$, $\cot\theta = -1$. 3. $(-4,-3)$: $r = 5$. $\sin\theta = -\frac{3}{5}$, $\cos\theta = -\frac{4}{5}$, $\tan\theta = \frac{3}{4}$, $\csc\theta = -\frac{5}{3}$, $\sec\theta = -\frac{5}{4}$, $\cot\theta = \frac{4}{3}$. 4. $(2,0)$: $r = 2$. $\sin\theta = 0$, $\cos\theta = 1$, $\tan\theta = 0$, $\csc\theta = \text{undefined}$, $\sec\theta = 1$, $\cot\theta = \text{undefined}$. 5. $(1,-8)$: $r = \sqrt{65}$. $\sin\theta = -\frac{8\sqrt{65}}{65}$, $\cos\theta = \frac{\sqrt{65}}{65}$, $\tan\theta = -8$, $\csc\theta = -\frac{\sqrt{65}}{8}$, $\sec\theta = \sqrt{65}$, $\cot\theta = -\frac{1}{8}$. 6. $(5,-3)$: $r = \sqrt{34}$. $\sin\theta = -\frac{3\sqrt{34}}{34}$, $\cos\theta = \frac{5\sqrt{34}}{34}$, $\tan\theta = -\frac{3}{5}$, $\csc\theta = -\frac{\sqrt{34}}{3}, \sec\theta = \frac{\sqrt{34}}{5}, \cot\theta = -\frac{5}{3}$. 7. $(-8,15)$: $r = 17$. $\sin\theta = \frac{15}{17}$, $\cos\theta = -\frac{8}{17}$, $\tan\theta = -\frac{15}{8}$, $\csc\theta = \frac{17}{15}$, $\sec\theta = -\frac{17}{8}$, $\cot\theta = -\frac{8}{15}$. 8. $(-1,-2)$: $r = \sqrt{5}$. $\sin\theta = -\frac{2\sqrt{5}}{5}$, $\cos\theta = -\frac{\sqrt{5}}{5}$, $\tan\theta = 2$, $\csc\theta = -\frac{\sqrt{5}}{2}$, $\sec\theta = -\sqrt{5}$, $\cot\theta = \frac{1}{2}$. ## Evaluate exact values for 9 to 16 We evaluate each trigonometric function at the given quadrantal angle: 9. $\sin\frac{\pi}{2} = 1$ 10. $\tan 2\pi = 0$ 11. $\cot(-180^\circ) = \text{undefined}$ (since $\sin(-180^\circ) = 0$) 12. $\csc 270^\circ = \frac{1}{\sin 270^\circ} = -1$ 13. $\cos(-270^\circ) = \cos 90^\circ = 0$ 14. $\sec 180^\circ = \frac{1}{\cos 180^\circ} = -1$ 15. $\tan\pi = 0$ 16. $\sec(-\frac{\pi}{2}) = \text{undefined}$ (since $\cos(-\frac{\pi}{2}) = 0$) ## Determine reference angles for 17 to 24 Using the *Radian Measure* and *Degree Measure* knowledge points, we find the reference angle $\theta'$ (the acute angle formed with the x-axis): 17. $135^\circ$ (Quadrant II): $\theta' = 180^\circ - 135^\circ = 45^\circ$ 18. $210^\circ$ (Quadrant III): $\theta' = 210^\circ - 180^\circ = 30^\circ$ 19. $\frac{7\pi}{12}$ (Quadrant II): $\theta' = \pi - \frac{7\pi}{12} = \frac{5\pi}{12}$ 20. $\frac{11\pi}{3}$ is coterminal with $\frac{5\pi}{3}$ (Quadrant IV): $\theta' = 2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$ 21. $-405^\circ$ is coterminal with $315^\circ$ (Quadrant IV): $\theta' = 360^\circ - 315^\circ = 45^\circ$ 22. $-75^\circ$ is coterminal with $285^\circ$ (Quadrant IV): $\theta' = 360^\circ - 285^\circ = 75^\circ$ 23. $\frac{5\pi}{6}$ (Quadrant II): $\theta' = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$ 24. $\frac{13\pi}{6}$ is coterminal with $\frac{\pi}{6}$ (Quadrant I): $\theta' = \frac{\pi}{6}$ | No. | Answer | |---|---| | 1 | $\sin\theta = \frac{4}{5}$, $\cos\theta = \frac{3}{5}$, $\tan\theta = \frac{4}{3}$, $\csc\theta = \frac{5}{4}$, $\sec\theta = \frac{5}{3}$, $\cot\theta = \frac{3}{4}$ | | 2 | $\sin\theta = \frac{\sqrt{2}}{2}$, $\cos\theta = -\frac{\sqrt{2}}{2}$, $\tan\theta = -1$, $\csc\theta = \sqrt{2}$, $\sec\theta = -\sqrt{2}$, $\cot\theta = -1$ | | 3 | $\sin\theta = -\frac{3}{5}$, $\cos\theta = -\frac{4}{5}$, $\tan\theta = \frac{3}{4}$, $\csc\theta = -\frac{5}{3}$, $\sec\theta = -\frac{5}{4}$, $\cot\theta = \frac{4}{3}$ | | 4 | $\sin\theta = 0$, $\cos\theta = 1$, $\tan\theta = 0$, $\csc\theta = \text{undefined}$, $\sec\theta = 1$, $\cot\theta = \text{undefined}$ | | 5 | $\sin\theta = -\frac{8\sqrt{65}}{65}$, $\cos\theta = \frac{\sqrt{65}}{65}$, $\tan\theta = -8$, $\csc\theta = -\frac{\sqrt{65}}{8}$, $\sec\theta = \sqrt{65}$, $\cot\theta = -\frac{1}{8}$ | | 6 | $\sin\theta = -\frac{3\sqrt{34}}{34}$, $\cos\theta = \frac{5\sqrt{34}}{34}$, $\tan\theta = -\frac{3}{5}$, $\csc\theta = -\frac{\sqrt{34}}{3}$, $\sec\theta = \frac{\sqrt{34}}{5}$, $\cot\theta = -\frac{5}{3}$ | | 7 | $\sin\theta = \frac{15}{17}$, $\cos\theta = -\frac{8}{17}$, $\tan\theta = -\frac{15}{8}$, $\csc\theta = \frac{17}{15}$, $\sec\theta = -\frac{17}{8}$, $\cot\theta = -\frac{8}{15}$ | | 8 | $\sin\theta = -\frac{2\sqrt{5}}{5}$, $\cos\theta = -\frac{\sqrt{5}}{5}$, $\tan\theta = 2$, $\csc\theta = -\frac{\sqrt{5}}{2}$, $\sec\theta = -\sqrt{5}$, $\cot\theta = \frac{1}{2}$ | | 9 | $1$ | | 10 | $0$ | | 11 | $\text{undefined}$ | | 12 | $-1$ | | 13 | $0$ | | 14 | $-1$ | | 15 | $0$ | | 16 | $\text{undefined}$ | | 17 | $45^\circ$ | | 18 | $30^\circ$ | | 19 | $\frac{5\pi}{12}$ | | 20 | $\frac{\pi}{3}$ | | 21 | $45^\circ$ | | 22 | $75^\circ$ | | 23 | $\frac{\pi}{6}$ | | 24 | $\frac{\pi}{6}$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios" ] }

No.	Answer
2	\(\sin\theta = \frac{\sqrt{2}}{2}\), \(\cos\theta = -\frac{\sqrt{2}}{2}\), \(\tan\theta = -1\), \(\csc\theta = \sqrt{2}\), \(\sec\theta = -\sqrt{2}\), \(\cot\theta = -1\)
3	\(\sin\theta = -\frac{3}{5}\), \(\cos\theta = -\frac{4}{5}\), \(\tan\theta = \frac{3}{4}\), \(\csc\theta = -\frac{5}{3}\), \(\sec\theta = -\frac{5}{4}\), \(\cot\theta = \frac{4}{3}\)
4	\(\sin\theta = 0\), \(\cos\theta = 1\), \(\tan\theta = 0\), \(\csc\theta = \text{undefined}\), \(\sec\theta = 1\), \(\cot\theta = \text{undefined}\)
5	\(\sin\theta = -\frac{8\sqrt{65}}{65}\), \(\cos\theta = \frac{\sqrt{65}}{65}\), \(\tan\theta = -8\), \(\csc\theta = -\frac{\sqrt{65}}{8}\), \(\sec\theta = \sqrt{65}\), \(\cot\theta = -\frac{1}{8}\)
6	\(\sin\theta = -\frac{3\sqrt{34}}{34}\), \(\cos\theta = \frac{5\sqrt{34}}{34}\), \(\tan\theta = -\frac{3}{5}\), \(\csc\theta = -\frac{\sqrt{34}}{3}\), \(\sec\theta = \frac{\sqrt{34}}{5}\), \(\cot\theta = -\frac{5}{3}\)
7	\(\sin\theta = \frac{15}{17}\), \(\cos\theta = -\frac{8}{17}\), \(\tan\theta = -\frac{15}{8}\), \(\csc\theta = \frac{17}{15}\), \(\sec\theta = -\frac{17}{8}\), \(\cot\theta = -\frac{8}{15}\)
8	\(\sin\theta = -\frac{2\sqrt{5}}{5}\), \(\cos\theta = -\frac{\sqrt{5}}{5}\), \(\tan\theta = 2\), \(\csc\theta = -\frac{\sqrt{5}}{2}\), \(\sec\theta = -\sqrt{5}\), \(\cot\theta = \frac{1}{2}\)
9	\(1\)
10	\(0\)
11	\(\text{undefined}\)
12	\(-1\)
13	\(0\)
14	\(-1\)
15	\(0\)
16	\(\text{undefined}\)
17	\(45^\circ\)
18	\(30^\circ\)
19	\(\frac{5\pi}{12}\)
20	\(\frac{\pi}{3}\)
21	\(45^\circ\)
22	\(75^\circ\)
23	\(\frac{\pi}{6}\)
24	\(\frac{\pi}{6}\)

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QUESTION IMAGE

Question

Explanation:

Find trigonometric functions for points 1 to 8

Evaluate exact values for 9 to 16

Determine reference angles for 17 to 24

Answer: