Inverse Trigonometric Functions
A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.
Inverse trig functions answer 'which angle has this trig value?' arcsin (also written sin^-1) undoes sin; arccos undoes cos; arctan undoes tan. To make the inverse a function, each one returns only the PRINCIPAL value: arcsin in [-90, 90], arccos in [0, 180], arctan in (-90, 90) (degrees).
What you'll learn
- Evaluate arcsin, arccos, and arctan for common input values
- Know the principal-value ranges: arcsin in [-90, 90], arccos in [0, 180], arctan in (-90, 90) degrees
- Use inverse trig to solve simple right-triangle problems
Why it matters: Robotics (joint angle from x/y reach), surveying (angle from height/distance), physics (launch angle from range and speed), and computer graphics all use inverse trig.
Worked example
Problem. Evaluate arcsin(1/2) in degrees.
- Ask: sin of what angle equals 1/2?
- sin(30 degrees) = 1/2 -> arcsin(1/2) = 30 degrees.
Answer: 30
Practice problems
1. arcsin(0) in degrees.
Show solution
- Warm-up: First identify exactly what the question is asking: arcsin(0) in degrees.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- sin(0) = 0.
- Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.
Answer: 0
2. arcsin(1) in degrees.
Show solution
- Warm-up: First identify exactly what the question is asking: arcsin(1) in degrees.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- sin(90) = 1.
- Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.
Answer: 90
3. arcsin(1/2) in degrees.
Show solution
- Warm-up: First identify exactly what the question is asking: arcsin(1/2) in degrees.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- sin(30) = 1/2.
- Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.
Answer: 30
4. arccos(1) in degrees.
Show solution
- Core Practice: First identify exactly what the question is asking: arccos(1) in degrees.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- cos(0) = 1.
- Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.
Answer: 0
5. arccos(0) in degrees.
Show solution
- Core Practice: First identify exactly what the question is asking: arccos(0) in degrees.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- cos(90) = 0.
- Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.
Answer: 90
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