Let’s solve each problem step by step. These are inverse trigonometric functions — they ask: “What angle gives me this value?” We’ll use the unit circle and special angles (like 0°, 30°, 45°, 60°, 90°) to find exact answers without a calculator.
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1. sin⁻¹(½)
We want the angle whose sine is ½.
From memory: sin(30°) = ½ → in radians, that’s π/6.
Also, arcsin only returns angles between -π/2 and π/2 (Quadrants I and IV).
So answer:
π/6
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2. cos⁻¹(½)
Cosine of what angle is ½?
cos(60°) = ½ → π/3 radians.
Arccos returns angles from 0 to π (Quadrants I and II).
So answer:
π/3
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3. tan⁻¹(√3 / 3)
Simplify √3 / 3 = 1/√3
tan(30°) = 1/√3 → π/6 radians.
Arctan returns angles from -π/2 to π/2.
So answer:
π/6
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4. arccos(√3 / 2)
cos(30°) = √3 / 2 → π/6
Arccos range: [0, π] → so it’s valid.
Answer:
π/6
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5. arcsin(√2 / 2)
sin(45°) = √2 / 2 → π/4
Arcsin range: [-π/2, π/2] → OK.
Answer:
π/4
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6. arctan(1)
tan(45°) = 1 → π/4
Arctan range: (-π/2, π/2) → OK.
Answer:
π/4
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7. arcsin(-½)
Negative input → angle in Quadrant IV.
sin(30°) = ½ → so sin(-30°) = -½ → -π/6
Answer:
-π/6
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8. arccos(-½)
Cosine negative → Quadrant II.
cos(60°) = ½ → so cos(120°) = -½ → 2π/3
Arccos range includes up to π → OK.
Answer:
2π/3
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9. arctan(-√3 / 3)
= arctan(-1/√3)
tan(30°) = 1/√3 → so tan(-30°) = -1/√3 → -π/6
Answer:
-π/6
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10. cos⁻¹(-√3 / 2)
Cosine negative → Quadrant II.
cos(30°) = √3 / 2 → so cos(150°) = -√3 / 2 → 5π/6
Answer:
5π/6
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11. sin⁻¹(-√2 / 2)
Negative → Quadrant IV.
sin(45°) = √2 / 2 → sin(-45°) = -√2 / 2 → -π/4
Answer:
-π/4
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12. tan⁻¹(-1)
tan(45°) = 1 → tan(-45°) = -1 → -π/4
Answer:
-π/4
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13. sin⁻¹(0)
sin(0) = 0 → angle is 0
Answer:
0
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14. cos⁻¹(0)
cos(90°) = 0 → π/2
Answer:
π/2
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15. tan⁻¹(-√3)
tan(60°) = √3 → tan(-60°) = -√3 → -π/3
Answer:
-π/3
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16. arcsin(1)
sin(90°) = 1 → π/2
Answer:
π/2
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17. arccos(1)
cos(0) = 1 → 0
Answer:
0
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18. tan⁻¹(0)
tan(0) = 0 → 0
Answer:
0
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19. arcsin(-1)
sin(-90°) = -1 → -π/2
Answer:
-π/2
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20. arccos(-1)
cos(180°) = -1 → π
Answer:
π
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Final Answer:
1. π/6
2. π/3
3. π/6
4. π/6
5. π/4
6. π/4
7. -π/6
8. 2π/3
9. -π/6
10. 5π/6
11. -π/4
12. -π/4
13. 0
14. π/2
15. -π/3
16. π/2
17. 0
18. 0
19. -π/2
20. π
Parent Tip: Review the logic above to help your child master the concept of inverse trig worksheet.
