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Worksheet 18 key for Inverse Trigonometric Functions, Exercises of Algebra

Ma110 Algebra and Trigonometry for Calculus at University of Kentucky (UK)

Typology: Exercises

2020/2021

Uploaded on 04/20/2021

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M110 Fa17 Page 1/7
Worksheet 18 KEY - Inverse Trigonometric Functions (§7.4)
1. arcsin (1) = π
22. arcsin 3
2!=π
33. arcsin 2
2!=π
4
4. arcsin 1
2=π
65. arcsin (0) = 0 6. arcsin 1
2=π
6
7. arcsin 2
2!=π
48. arcsin 3
2!=π
39. arcsin (1) = π
2
10. arccos (1) = π11. arccos 3
2!=5π
612. arccos 2
2!=3π
4
13. arccos 1
2=2π
314. arccos (0) = π
215. arccos 1
2=π
3
16. arccos 2
2!=π
417. arccos 3
2!=π
618. arccos (1) = 0
19. arctan "3=π
320. arctan (1) = π
421. arctan 3
3!=π
6
22. arctan (0) = 0 23. arctan 3
3!=π
624. arctan (1) = π
4
25. arctan "3=π
326. arccot "3=5π
627. arccot (1) = 3π
4
28. arccot 3
3!=2π
329. arccot (0) = π
230. arccot 3
3!=π
3
31. arccot (1) = π
432. arccot "3=π
633. arcsec (2) = π
3
34. arccsc (2) = π
635. arcsec "2=π
436. arccsc "2=π
4
37. arcsec 23
3!=π
638. arccsc 23
3!=π
339. arcsec (1) = 0
40. arccsc (1) = π
241. arcsec (2) = 4π
342. arcsec "2=5π
4
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Worksheet 18 KEY - Inverse Trigonometric Functions (§7. 4 )

  1. arcsin (−1) = −

π

2

  1. arcsin

π

3

  1. arcsin

π

4

  1. arcsin

π

6

  1. arcsin (0) = 0 6. arcsin

π

6

  1. arcsin

π

4

  1. arcsin

π

3

  1. arcsin (1) =

π

2

  1. arccos (−1) = π 11. arccos

5 π

6

  1. arccos

3 π

4

  1. arccos

2 π

3

  1. arccos (0) =

π

2

  1. arccos

π

3

  1. arccos

π

4

  1. arccos

π

6

  1. arccos (1) = 0
  2. arctan

π

3

  1. arctan (−1) = −

π

4

  1. arctan

π

6

  1. arctan (0) = 0 23. arctan

π

6

  1. arctan (1) =

π

4

  1. arctan

π

3

  1. arccot

5 π

6

  1. arccot (−1) =

3 π

4

  1. arccot

2 π

3

  1. arccot (0) =

π

2

  1. arccot

π

3

  1. arccot (1) =

π

4

  1. arccot

π

6

  1. arcsec (2) =

π

3

  1. arccsc (2) =

π

6

  1. arcsec

π

4

  1. arccsc

π

4

  1. arcsec

π

6

  1. arccsc

π

3

  1. arcsec (1) = 0
  2. arccsc (1) =

π

2

  1. arcsec (−2) =

4 π

3

  1. arcsec

5 π

4

  1. arcsec

7 π

6

  1. arcsec (−1) = π 45. arccsc (−2) =

7 π

6

  1. arccsc

5 π

4

  1. arccsc

4 π

3

  1. arccsc (−1) =

3 π

2

  1. arcsec (−2) =

2 π

3

  1. arcsec

3 π

4

  1. arcsec

5 π

6

  1. arcsec (−1) = π 53. arccsc (−2) = −

π

6

  1. arccsc

π

4

  1. arccsc

π

3

  1. arccsc (−1) = −

π

2

  1. sin

arcsin

  1. sin

arcsin

  1. sin

arcsin

  1. sin (arcsin (− 0 .42)) = − 0. 42
  2. sin

arcsin

is undefined. 62. cos

arccos

  1. cos

arccos

  1. cos

arccos

  1. cos (arccos (− 0 .998)) = − 0. 998 66. cos (arccos (π)) is undefined.
  2. tan (arctan (−1)) = − 1 68. tan

arctan

  1. tan

arctan

  1. tan (arctan (0.965)) = 0. 965
  2. tan (arctan (3π)) = 3π 72. cot (arccot (1)) = 1
  3. cot

arccot

3 74. cot

arccot

  1. cot (arccot (− 0 .001)) = − 0. 001 76. cot

arccot

17 π

4

17 π

4

  1. sec (arcsec (2)) = 2 78. sec (arcsec (−1)) = − 1
  1. arccsc

csc

5 π

4

5 π

4

  1. arccsc

csc

2 π

3

π

3

  1. arccsc

csc

π

2

3 π

2

  1. arccsc

csc

11 π

6

7 π

6

  1. arcsec

sec

11 π

12

13 π

12

  1. arccsc

csc

9 π

8

9 π

8

  1. arcsec

sec

π

4

π

4

  1. arcsec

sec

4 π

3

2 π

3

  1. arcsec

sec

5 π

6

5 π

6

  1. arcsec

sec

π

2

is undefined.

  1. arcsec

sec

5 π

3

π

3

  1. arccsc

csc

( (^) π

π

6

  1. arccsc

csc

5 π

4

π

4

  1. arccsc

csc

2 π

3

π

3

  1. arccsc

csc

π

2

π

2

  1. arccsc

csc

11 π

6

π

6

  1. arcsec

sec

11 π

12

11 π

12

  1. arccsc

csc

9 π

8

π

8

  1. sin

arccos

  1. sin

arccos

  1. sin (arctan (−2)) = −
  1. sin

arccot

  1. sin (arccsc (−3)) = −
  1. cos

arcsin

  1. cos

arctan

  1. cos (arccot (3)) =
  1. cos (arcsec (5)) =
  1. tan

arcsin

  1. tan

arccos

3 142. tan

arcsec

  1. tan (arccot (12)) =
  1. cot

arcsin

  1. cot

arccos

3 146. cot

arccsc

  1. cot (arctan (0.25)) = 4 148. sec

arccos

  1. sec

arcsin

  1. sec (arctan (10)) =
  1. sec

arccot

11 152. csc (arccot (9)) =

  1. csc

arcsin

  1. csc

arctan

  1. sin

arcsin

π

4

  1. cos (arcsec(3) + arctan(2)) =
  1. tan

arctan(3) + arccos

  1. sin

2 arcsin

  1. sin

2arccsc

  1. sin (2 arctan (2)) =
  1. cos

2 arcsin

  1. cos

2arcsec

  1. cos

2arccot

  1. sin

arctan(2)

2

  1. sin (arccos (x)) =

1 − x^2 for − 1 ≤ x ≤ 1

  1. cos (arctan (x)) =

1 + x^2

for all x

  1. tan (arcsin (x)) =

x √ 1 − x^2

for − 1 < x < 1

  1. sec (arctan (x)) =

1 + x^2 for all x

  1. csc (arccos (x)) =

1 − x^2

for − 1 < x < 1

  1. sin (2 arctan (x)) =

2 x

x^2 + 1

for all x

  1. sin (2 arccos (x)) = 2x

1 − x^2 for − 1 ≤ x ≤ 1

  1. If sec(θ) =

x

4

for 0 < θ <

π

2

, then 4 tan(θ) − 4 θ =

x^2 − 16 − 4arcsec

x

4

  1. x = arcsin
  • 2πk or x = π − arcsin
  • 2πk, in [0, 2 π), x ≈ 0. 6898 , 2. 4518
  1. x = arccos
  • 2πk or x = − arccos
  • 2πk, in [0, 2 π), x ≈ 1. 7949 , 4. 4883
  1. x = π + arcsin(0.569) + 2πk or x = 2π − arcsin(0.569) + 2πk, in [0, 2 π), x ≈ 3. 7469 , 5. 6779
  2. x = arccos(0.117) + 2πk or x = 2π − arccos(0.117) + 2πk, in [0, 2 π), x ≈ 1. 4535 , 4. 8297
  3. x = arcsin(0.008) + 2πk or x = π − arcsin(0.008) + 2πk, in [0, 2 π), x ≈ 0. 0080 , 3. 1336
  4. x = arccos
  • 2πk or x = 2π − arccos
  • 2πk, in [0, 2 π), x ≈ 0. 0746 , 6. 2086
  1. x = arctan(117) + πk, in [0, 2 π), x ≈ 1. 56225 , 4. 70384
  2. x = arctan
  • πk, in [0, 2 π), x ≈ 3. 0585 , 6. 2000
  1. x = arccos
  • 2πk or x = 2π − arccos
  • 2πk, in [0, 2 π), x ≈ 0. 8411 , 5. 4422
  1. x = π + arcsin
  • 2πk or x = 2π − arcsin
  • 2πk, in [0, 2 π), x ≈ 3. 3316 , 6. 0932
  1. x = arctan
  • πk, in [0, 2 π), x ≈ 1. 8771 , 5. 0187
  1. x = arcsin
  • 2πk or x = π − arcsin
  • 2πk, in [0, 2 π), x ≈ 0. 3844 , 2. 7572
  1. x = arccos
  • 2πk or x = − arccos
  • 2πk, in [0, 2 π), x ≈ 2. 0236 , 4. 2596
  1. x = arctan(0.03) + πk, in [0, 2 π), x ≈ 0. 0300 , 3. 1716
  2. x = arcsin(0.3502) + 2πk or x = π − arcsin(0.3502) + 2πk, in [0, 2 π), x ≈ 0. 3578 , 2. 784
  3. x = π + arcsin(0.721) + 2πk or x = 2π − arcsin(0.721) + 2πk, in [0, 2 π), x ≈ 3. 9468 , 5. 4780
  4. x = arccos(0.9824) + 2πk or x = 2π − arccos(0.9824) + 2πk, in [0, 2 π), x ≈ 0. 1879 , 6. 0953
  5. x = arccos(− 0 .5637) + 2πk or x = − arccos(− 0 .5637) + 2πk, in [0, 2 π), x ≈ 2. 1697 , 4. 1135
  6. x = arctan(117) + πk, in [0, 2 π), x ≈ 1. 5622 , 4. 7038
  7. x = arctan(− 0 .6109) + πk, in [0, 2 π), x ≈ 2. 5932 , 5. 7348

208. 68. 9 ◦^209. 7. 7 ◦^210. 51◦^211. 19. 5 ◦^212. 41. 81 ◦