Lesson 4: Inverse Trig Functions
Estimated time: 35-40 minutes
Learning Objectives
- Define arcsin, arccos, and arctan with restricted domains
- Evaluate inverse trig functions at exact values
- Compose trig and inverse trig functions
- Understand the domain and range of each inverse function
Why We Need Restricted Domains
Trig functions are not one-to-one (they fail the horizontal line test). To define an inverse, we restrict the domain so each output corresponds to exactly one input.
The Three Main Inverse Functions
y = sin¹(x) = arcsin(x)
Domain: [−1, 1] | Range: [−π/2, π/2]
Returns the angle in [−π/2, π/2] whose sine is x.
y = cos¹(x) = arccos(x)
Domain: [−1, 1] | Range: [0, π]
Returns the angle in [0, π] whose cosine is x.
y = tan¹(x) = arctan(x)
Domain: (−∞, ∞) | Range: (−π/2, π/2)
Returns the angle in (−π/2, π/2) whose tangent is x.
Evaluating Inverse Functions
Example 1: arcsin(1/2)
What angle in [−π/2, π/2] has sine = 1/2? Answer: π/6.
Example 2: arccos(−√2/2)
What angle in [0, π] has cosine = −√2/2? Answer: 3π/4.
Example 3: arctan(1)
What angle in (−π/2, π/2) has tangent = 1? Answer: π/4.
Example 4: arcsin(−1)
What angle in [−π/2, π/2] has sine = −1? Answer: −π/2.
Compositions
sin(arcsin x) = x for x in [−1, 1]
arcsin(sin x) = x only when x is in [−π/2, π/2]
Example 5: Composition
Find cos(arcsin(3/5)).
Let θ = arcsin(3/5), so sinθ = 3/5 and θ in QI. cosθ = 4/5. Answer: 4/5.
Example 6: arcsin(sin(5π/6))
sin(5π/6) = 1/2. arcsin(1/2) = π/6. Not 5π/6 because 5π/6 is outside [−π/2, π/2]. Answer: π/6.
Check Your Understanding
1. Find arccos(0).
2. Find arctan(−√3).
3. Find sin(arccos(5/13)).
Key Takeaways
- arcsin returns angles in [−π/2, π/2], arccos in [0, π], arctan in (−π/2, π/2).
- Always check that the answer is in the correct restricted range.
- For compositions like cos(arcsin x), draw a right triangle to find the answer.
- Inverse trig functions are essential for solving trig equations.