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Precalculus Unit 8: Inverse trig & trig equations

Inverse trig functions undo sine, cosine and tangent on restricted domains. They let you solve trig equations for an angle.

Covers: Inverse trig functions, solving trig equations & inequalities

Study guide

Inverse trig functions undo sine, cosine and tangent, but only on restricted domains, so each gives a single answer. arcsin x (sin⁻¹ x) returns an angle in [−π/2, π/2], arccos x returns an angle in [0, π], and arctan x returns an angle in (−π/2, π/2). For example, arcsin(1/2) = π/6. Note that sin⁻¹ x means the inverse function, not 1/sin x.

Solving a trig equation means finding all angles that work, not just one. Isolate the trig function, find the reference angle, and use the quadrant signs to find every solution in [0, 2π). For instance, sin x = 1/2 gives x = π/6 and 5π/6. To give all solutions, add multiples of the period: 2πk for sine and cosine, πk for tangent.

Equations may need algebra first: factor (2sin²x − sin x = 0 becomes sin x(2sin x − 1) = 0), use identities to rewrite everything in one function, or substitute for a multiple angle (for sin 2x = 1, solve for 2x first, then divide). Don't divide both sides by a trig function, which can lose solutions.

Trig inequalities are solved by finding the solutions of the related equation, which split the interval into pieces, then using the graph or unit circle to see where the inequality holds.

arcsin(sin x) = x only when x is in [−π/2, π/2], and arccos(cos x) = x only when x is in [0, π]. Otherwise, find the angle in the inverse function's range that has the same sine (or cosine) value, as in arccos(cos(5π/3)) = π/3. To evaluate something like cos(arcsin(3/5)), draw a right triangle with an angle whose sine is 3/5, use the Pythagorean theorem to get the third side (4), and read off cos = 4/5. Choose the sign from the quadrant allowed by the inverse function's range.

arcsin(−x) = −arcsin x and arctan(−x) = −arctan x, but arccos(−x) = π − arccos x. Also, arcsin x + arccos x = π/2.

Key ideas

• arcsin returns angles in [−π/2, π/2]; arccos in [0, π]; arctan in (−π/2, π/2).
• A trig equation usually has infinitely many solutions; add multiples of the period.
• Solve on [0, 2π) first, then generalize.
• Isolate the trig function, find the reference angle, then use quadrant signs.

Formulas and rules

Common mistakes

• Giving only one solution when the interval contains two.
• Thinking sin⁻¹x means 1/sin x.

Words to know

Range of arccos

[0, π]

arcsin(1/2)

π/6

Solving sin x = k on [0, 2π)

Usually two solutions: the reference angle in two quadrants.

Range of arctan

(−π/2, π/2)

General solution (tangent)

Add πk: tan x = 1 → x = π/4 + πk.

Range of arcsin

[−π/2, π/2]

General solution (sine and cosine)

Add 2πk to each solution in [0, 2π).

arcsin x

The angle in [−π/2, π/2] whose sine is x.

arccos x

The angle in [0, π] whose cosine is x.

arctan x

The angle in (−π/2, π/2) whose tangent is x.

Trig equation

An equation with trig functions; find every solution in the interval.

Trig inequality

Solve the related equation, then test the intervals between solutions.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.