You’ve uploaded a worksheet titled
“Precalculus Worksheet — Section 6.7 — Inverse Trig Functions”, which asks students to
evaluate inverse trigonometric expressions without using a calculator.
Let’s solve each problem step-by-step and explain the reasoning. The key is understanding the
domains and ranges of inverse trig functions, and knowing the
exact values of sine, cosine, and tangent for common angles (like 0°, 30°, 45°, 60°, 90°, etc., or in radians: 0, π/6, π/4, π/3, π/2).
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🔁 Inverse Trig Function Basics:
-
arcsin(x) = sin⁻¹(x): Range is
[-π/2, π/2] → outputs angles in QI or QIV
-
arccos(x) = cos⁻¹(x): Range is
[0, π] → outputs angles in QI or QII
-
arctan(x) = tan⁻¹(x): Range is
(-π/2, π/2) → outputs angles in QI or QIV
Also:
- sin(π/6) = 1/2 → arcsin(1/2) = π/6
- cos(π/3) = 1/2 → arccos(1/2) = π/3
- tan(π/4) = 1 → arctan(1) = π/4
- etc.
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##
✔ Solutions with Explanations:
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1. arcsin(√2/2)
→ sin(π/4) = √2/2, and π/4 is in [-π/2, π/2]
✔ Answer: π/4
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2. arccos(√2/2)
→ cos(π/4) = √2/2, and π/4 is in [0, π]
✔ Answer: π/4
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3. arcsin(1/2)
→ sin(π/6) = 1/2, and π/6 ∈ [-π/2, π/2]
✔ Answer: π/6
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4. arccos(-1/2)
→ cos(2π/3) = -1/2, and 2π/3 ∈ [0, π]
✔ Answer: 2π/3
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5. arcsin(-1/2)
→ sin(-π/6) = -1/2, and -π/6 ∈ [-π/2, π/2]
✔ Answer: -π/6
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6. arccos(0)
→ cos(π/2) = 0, and π/2 ∈ [0, π]
✔ Answer: π/2
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7. arcsin(-√2/2)
→ sin(-π/4) = -√2/2, and -π/4 ∈ [-π/2, π/2]
✔ Answer: -π/4
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8. arccos(-√2/2)
→ cos(3π/4) = -√2/2, and 3π/4 ∈ [0, π]
✔ Answer: 3π/4
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9. arcsin(√3/2)
→ sin(π/3) = √3/2, and π/3 ∈ [-π/2, π/2]
✔ Answer: π/3
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10. arccos(-√3/2)
→ cos(5π/6) = -√3/2, and 5π/6 ∈ [0, π]
✔ Answer: 5π/6
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11. arcsin(-√3/2)
→ sin(-π/3) = -√3/2, and -π/3 ∈ [-π/2, π/2]
✔ Answer: -π/3
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12. arctan(0)
→ tan(0) = 0, and 0 ∈ (-π/2, π/2)
✔ Answer: 0
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13. arcsin(0)
→ sin(0) = 0, and 0 ∈ [-π/2, π/2]
✔ Answer: 0
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14. arccos(0) ← Already done in #6
✔ Answer: π/2
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15. arctan(√3)
→ tan(π/3) = √3, and π/3 ∈ (-π/2, π/2)
✔ Answer: π/3
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16. arccos(1)
→ cos(0) = 1, and 0 ∈ [0, π]
✔ Answer: 0
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17. arcsin(1)
→ sin(π/2) = 1, and π/2 ∈ [-π/2, π/2]
✔ Answer: π/2
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18. arctan(-1)
→ tan(-π/4) = -1, and -π/4 ∈ (-π/2, π/2)
✔ Answer: -π/4
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19. arccos(-1)
→ cos(π) = -1, and π ∈ [0, π]
✔ Answer: π
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20. arcsin(0) ← Already done in #13
✔ Answer: 0
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## 📝 Final Answers List:
1.
π/4
2.
π/4
3.
π/6
4.
2π/3
5.
-π/6
6.
π/2
7.
-π/4
8.
3π/4
9.
π/3
10.
5π/6
11.
-π/3
12.
0
13.
0
14.
π/2
15.
π/3
16.
0
17.
π/2
18.
-π/4
19.
π
20.
0
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## 💡 Tips for Success:
- Memorize the unit circle values for sine, cosine, and tangent at standard angles.
- Always check that your answer falls within the
range of the inverse function.
- Negative inputs often give negative angles for arcsin and arctan, but positive angles for arccos (since its range is [0, π]).
Let me know if you want a printable version or flashcards for these! 😊
Parent Tip: Review the logic above to help your child master the concept of inverse trig worksheet.
