Inverse Trig Functions — Trigonometry Unit 6 practice.
This unit covers arcsin arccos arctan, evaluating inverse trig and compositions of inverse trig — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers arcsin arccos arctan, evaluating inverse trig and compositions of inverse trig — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Arcsin, Arccos, Arctan
Inverse trig functions return the angle whose trig value equals the input. Each has a restricted domain and range that you must memorize: arcsin and arctan output angles in [-π/2, π/2], while arccos outputs angles in [0, π]. These restrictions ensure the inverse is a true function.
Key Points
- arcsin(x) is defined for x ∈ [-1, 1], outputs angles in [-π/2, π/2]
- arccos(x) is defined for x ∈ [-1, 1], outputs angles in [0, π]
- arctan(x) is defined for all real x, outputs angles in (-π/2, π/2)
- The output of an inverse trig function is always an ANGLE, not a ratio
State the domain and range of f(x) = arccos(x).
The input x must satisfy -1 ≤ x ≤ 1, so the domain is [-1, 1]. The output is a restricted angle from the cosine function, so the range is [0, π]. Note that arccos never outputs a negative angle, which distinguishes it from arcsin.
2 Evaluating Inverse Trig
To evaluate an inverse trig expression, ask 'what angle in the restricted range has this trig value?' You must know exact values from the unit circle for the standard angles (0, π/6, π/4, π/3, π/2). Answers must stay within the function's restricted range.
Key Points
- arcsin(1/2) = π/6 because sin(π/6) = 1/2 and π/6 is in [-π/2, π/2]
- arccos(−1/2) = 2π/3 because cos(2π/3) = −1/2 and 2π/3 is in [0, π]
- arctan(−1) = −π/4 because tan(−π/4) = −1 and −π/4 is in (−π/2, π/2)
- Never give an answer outside the restricted range — e.g., arcsin(1/2) ≠ 5π/6
Evaluate arctan(√3).
You need the angle θ in (−π/2, π/2) such that tan(θ) = √3. From the unit circle, tan(π/3) = √3, and π/3 is within the restricted range. Therefore arctan(√3) = π/3.
3 Compositions Of Inverse Trig
When trig and inverse trig functions are composed, they do NOT always cancel. They cancel cleanly only when the input is within the restricted range of the inverse function. Outside that range, you must use the restricted-range output and re-evaluate.
Key Points
- sin(arcsin(x)) = x for all x ∈ [-1, 1] — these always cancel
- arcsin(sin(x)) = x ONLY if x ∈ [-π/2, π/2]; otherwise find the equivalent angle in that range
- For compositions like cos(arctan(x)), draw a right triangle: label sides using the definition of arctan, then find the cosine of that triangle
- Right triangle method: if arctan(3/4) = θ, then opposite = 3, adjacent = 4, hypotenuse = 5
Evaluate cos(arctan(3/4)).
Let θ = arctan(3/4), meaning tan(θ) = 3/4 with θ in (−π/2, π/2). Draw a right triangle with opposite side 3 and adjacent side 4; the hypotenuse is √(9+16) = 5. Since θ is in the first quadrant (positive value), cos(θ) = adjacent/hypotenuse = 4/5.
Questions, answered.
What is Inverse Trig Functions?
Inverse Trig Functions is Unit 6 of Trigonometry, covering arcsin arccos arctan, evaluating inverse trig and compositions of inverse trig.
How to study for Trigonometry Unit 6?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.