6.3Understanding Inverse Trigonometric Functions
Define arcsin (domain [−1,1], range [−π/2, π/2]), arccos (range [0, π]), and arctan (range (−π/2, π/2)). Evaluate exact values and compose trig with inverse trig functions.
Inverse trig functions are used to find angles from ratios — essential in physics, navigation, and engineering. In calculus, arcsin, arccos, and arctan have their own derivative and integral formulas that appear constantly.
Essential Question
Why must the domain of sine, cosine, and tangent be restricted before defining their inverses, and what are the ranges of arcsin, arccos, and arctan?
Lesson Overview
Why Restrict the Domain
A function must be one-to-one (pass the HLT) to have an inverse. Sine, cosine, and tangent are periodic — they repeat values — so we restrict each to a standard interval.
arcsin(x)
Inverse of sin on [−π/2, π/2]. Domain: [−1, 1]. Range: [−π/2, π/2]. arcsin(x) gives the angle whose sine is x.
arccos(x)
Inverse of cos on [0, π]. Domain: [−1, 1]. Range: [0, π]. arccos(x) gives the angle whose cosine is x.
arctan(x)
Inverse of tan on (−π/2, π/2). Domain: all reals. Range: (−π/2, π/2). Horizontal asymptotes at y = ±π/2.
Restricted Domain — Horizontal Line Test
Reference Table
y = arcsin(x)
y = arccos(x)
y = arctan(x)
Key Vocabulary
Inverse function
A function that reverses the input/output of another function; f⁻¹(f(x)) = x.
One-to-one function
A function where each output corresponds to exactly one input; passes the horizontal line test.
Horizontal line test
A visual test: if every horizontal line crosses the graph at most once, the function is one-to-one.
arcsin (inverse sine)
arcsin(x) = the angle θ in [−π/2, π/2] such that sin(θ) = x.
arccos (inverse cosine)
arccos(x) = the angle θ in [0, π] such that cos(θ) = x.
arctan (inverse tangent)
arctan(x) = the angle θ in (−π/2, π/2) such that tan(θ) = x.
Worked Examples
Evaluate arcsin(√2/2). Give an exact answer in radians.
We need the angle θ in [−π/2, π/2] such that sin(θ) = √2/2.
From the unit circle, sin(π/4) = √2/2.
π/4 is in [−π/2, π/2], so it is the valid answer.
Evaluate arccos(−1/2). Give an exact answer in radians.
We need the angle θ in [0, π] such that cos(θ) = −1/2.
cos(2π/3) = −1/2, and 2π/3 is in [0, π].
Evaluate arctan(−1). Give an exact answer in radians.
We need θ in (−π/2, π/2) such that tan(θ) = −1.
tan(−π/4) = −1, and −π/4 is in (−π/2, π/2).
Simplify sin(arccos(3/5)) without a calculator.
Let θ = arccos(3/5), so cos(θ) = 3/5 and θ is in [0, π].
Draw a right triangle: adjacent = 3, hypotenuse = 5.
By the Pythagorean theorem: opposite = √(25 − 9) = √16 = 4.
sin(θ) = opposite/hypotenuse = 4/5.
Simplify cos(arctan(x)) and express in terms of x.
Let θ = arctan(x), so tan(θ) = x/1.
Draw a right triangle: opposite = x, adjacent = 1, hypotenuse = √(x² + 1).
cos(θ) = adjacent/hypotenuse = 1/√(x² + 1).
Guided Practice
Evaluate arcsin(−1/2).
Hint: Find θ in [−π/2, π/2] with sin(θ) = −1/2. Think about the 30-60-90 triangle.
Evaluate arccos(0).
Hint: Find θ in [0, π] with cos(θ) = 0. Which standard angle has cosine 0?
Evaluate arctan(√3).
Hint: Find θ in (−π/2, π/2) with tan(θ) = √3. Use the 30-60-90 triangle.
Simplify cos(arcsin(5/13)).
Hint: Let θ = arcsin(5/13). Draw a right triangle with opposite = 5, hypotenuse = 13. Find the adjacent side.
Simplify tan(arcsin(x)).
Hint: Let θ = arcsin(x), so sin(θ) = x/1. Draw a right triangle and find tan(θ) in terms of x.
Interactive Practice — 5 Questions
What is arcsin(1)?
What is the range of arccos(x)?
Evaluate arctan(0).
Which value is NOT in the domain of arcsin(x)?
Simplify sin(arcsin(0.7)).
Common Mistakes
arcsin(2) = some angle
arcsin(x) is undefined for |x| > 1 — the domain is [−1, 1] only.
arccos(−1/2) = −π/3 (negative angle)
arccos always returns values in [0, π]. arccos(−1/2) = 2π/3, not −π/3.
sin(arcsin(x)) = x for all x
sin(arcsin(x)) = x only for x in [−1, 1]. Outside this domain, arcsin is undefined.
arcsin(sin(x)) = x for all x
arcsin(sin(x)) = x only when x is in [−π/2, π/2]. For other x, arcsin wraps back to the restricted range.
Math Tips
Memorize the three restricted ranges: arcsin → [−π/2, π/2], arccos → [0, π], arctan → (−π/2, π/2).
For right-triangle compositions like sin(arccos(x)), draw a triangle — label two sides, find the third with the Pythagorean theorem.
arctan has no domain restriction — it accepts any real number — but its range is still bounded by ±π/2.
The notation sin⁻¹(x) means arcsin(x), NOT 1/sin(x). The reciprocal of sin is csc, not sin⁻¹.
On the SAT/ACT, inverse trig questions usually ask for exact radian values from the unit circle — know your 30-60-90 and 45-45-90 triangles cold.