3.9 Inverse Trigonometric Functions

AP Precalculus 3.9 Inverse Trigonometric Functions Summary

Inverse trigonometric functions (arcsine, arccosine, and arctangent) reverse the sine, cosine, and tangent functions: you give them a ratio value and they return an angle. Because sine, cosine, and tangent are periodic, you have to restrict their domains first so each inverse gives one unique angle.

Inverse trigonometric functions take a sine, cosine, or tangent value and return an angle, and you may need to switch between a trig function and its inverse using graphs, tables, or equations before building on them to solve trigonometric equations in the next topic. Some questions are calculator-active and some are not, so recall exact-value cases like arcsin(1/2) without technology, use a calculator for the rest, and write clear notation with correct intervals.

Restricted Domains Make Trig Functions Invertible

• An inverse trig function swaps input and output: the input is a value in the range of the original function, and the output is an angle.
• The three inverses are arcsine, arccosine, and arctangent, also written as sin^(-1) x, cos^(-1) x, and tan^(-1) x.
• These inverses exist only because the original functions get restricted domains, making them one-to-one.
• Restricted domains: sine on [-π/2, π/2], cosine on [0, π], and tangent on (-π/2, π/2).
• The restricted domain of the original function becomes the range (the output angles) of its inverse.
• Each inverse graph is a reflection of the restricted original function across the line y = x.

Arcsine, Arccosine, and Arctangent

An inverse trig function reverses what its trig function does. A trig function takes an angle and gives a ratio value. The inverse takes that ratio value and gives back an angle. That is why the output of an inverse trig function is usually read as an angle measure, while the input is a value that the original function could output.

The three inverses are arcsine, arccosine, and arctangent. You will also see them written as sin^(-1) x, cos^(-1) x, and tan^(-1) x. The "-1" here means inverse function, not a reciprocal, so sin^(-1) x is not the same as 1/sin x.

There is one catch. Sine, cosine, and tangent are periodic, so they repeat the same output over and over. That means many different angles share the same sine value, the same cosine value, or the same tangent value. A function can only have an inverse if each input maps to exactly one output, so you cannot invert the full sine, cosine, or tangent function. To fix this, you restrict the domain of each original function to a stretch where it is one-to-one.

Restricting the Domain of the Sine Function

The sine function repeats every 2π radians, so for any sine value there are infinitely many angles that produce it. For example, sin(30°) equals sin(390°) because 30 + 360 = 390. With that many matches, there is no single angle to return.

To get one unique angle, sine is restricted to [-π/2, π/2]. On this interval sine increases steadily from -1 to 1 and never repeats, so each value is hit exactly once. This restricted domain becomes the output range of arcsine, which means arcsine returns angles only in [-π/2, π/2]. The square brackets show a closed interval, so -π/2 and π/2 are valid outputs.

Restricting the Domain of the Cosine Function

Cosine also repeats every 2π radians, so the same problem shows up: many angles share a cosine value. For example, cos(60°) equals cos(420°).

To make cosine invertible, its domain is restricted to [0, π]. On this stretch cosine decreases steadily from 1 to -1 without repeating, so each value matches one angle. That restricted domain becomes the range of arccosine, so arccosine returns angles in [0, π]. These are also square brackets, so the endpoints are included.

Restricting the Domain of the Tangent Function

Tangent repeats every π radians and is undefined wherever cosine is zero, such as at π/2 and -π/2. Its output covers all real numbers.

To define arctangent, tangent is restricted to (-π/2, π/2). On this interval tangent increases through every real number exactly once. That restricted domain becomes the range of arctangent, so arctangent returns angles in (-π/2, π/2). Notice the parentheses: this is an open interval, so -π/2 and π/2 are never outputs because tangent is undefined there. That is also why the arctangent graph has horizontal asymptotes at y = π/2 and y = -π/2 instead of stopping at endpoints.

Graphs of the Inverse Functions

Each inverse graph is the reflection of its restricted original function across the line y = x. Reflecting across y = x is exactly what happens when you swap inputs and outputs.

• Arcsine: domain [-1, 1], range [-π/2, π/2]. It increases from left to right.
• Arccosine: domain [-1, 1], range [0, π]. It decreases from left to right.
• Arctangent: domain all real numbers, range (-π/2, π/2). It increases and flattens toward the horizontal asymptotes y = ±π/2.

The arcsine and arccosine graphs are finite in both directions because their domains stop at -1 and 1. The arctangent graph keeps going left and right forever because its domain is every real number.

Using the Unit Circle to Evaluate Inverse Trig Functions

The unit circle is the fastest way to evaluate exact inverse values. The input of an inverse trig function is a sine, cosine, or tangent value, so the real question is: what angle (inside the allowed range) has that value?

Example: evaluate arctan(1).

Ask yourself: what angle has a tangent value of 1, staying inside (-π/2, π/2)? On the unit circle, tan(π/4) = 1, and π/4 is inside the allowed range, so arctan(1) = π/4.

The key step is to stay inside each inverse's range. There may be other angles with the same trig value, but only the one inside the restricted range is the correct answer.

Evaluating Inverses and Avoiding the arcsin(sin x) Trap

Problem Solving:

• Read the input as a trig value and the output as an angle. Then ask, "what angle gives me this value?"
• Keep your answer inside the correct range: arcsine in [-π/2, π/2], arccosine in [0, π], arctangent in (-π/2, π/2).
• Memorize the common exact values so you can evaluate arcsin, arccos, and arctan of values like 0, 1/2, √2/2, √3/2, and 1 without a calculator.

Common Trap:

• arcsin(sin x) does not always equal x. It only returns x when x is already inside [-π/2, π/2]. If x is outside that range, arcsine sends back the matching angle that is inside the range instead.
• Going the other direction, sin(arcsin x) = x for any x in [-1, 1], because that input is already a valid sine value.

Practice Problems: 1. What is the value of arcsin(-1/2)?

a) -30 degrees b) -60 degrees c) 30 degrees d) 60 degrees

Answer: a) -30 degrees. The angle in [-π/2, π/2] with sine -1/2 is -30 degrees.

2. What is the value of arccos(√2/2)?

a) 30 degrees b) 45 degrees c) 60 degrees d) 90 degrees

Answer: b) 45 degrees. The angle in [0, π] with cosine √2/2 is 45 degrees.

3. What is the value of arctan(√3)?

a) 45 degrees b) 60 degrees c) 90 degrees d) 120 degrees

Answer: b) 60 degrees. The angle in (-π/2, π/2) with tangent √3 is 60 degrees.

Common Misconceptions

• Misconception: sin^(-1) x is 1/sin x.
  Actually: sin^(-1) x is not 1/sin x. The "-1" means inverse function, not reciprocal. The reciprocal of sine is cosecant, which is a different idea.
• Misconception: Inverse trig functions return every possible angle.
  Actually: Inverse trig functions do not return every possible angle. Each one returns only the single angle inside its restricted range, even though many angles share the same trig value.
• Misconception: The endpoints are the same for every inverse.
  Actually: The endpoints are not the same for every inverse. Arcsine and arccosine include their range endpoints (closed intervals), but arctangent never reaches ±π/2 (open interval), because tangent is undefined there.
• Misconception: Arcsine and arccosine accept inputs outside [-1, 1].
  Actually: The input must be valid. Arcsine and arccosine only accept inputs in [-1, 1], since sine and cosine never output values outside that range. A value like arccos(2) has no answer.
• Misconception: Restricting the domain is optional.
  Actually: Restricting the domain is not optional. Without the restriction, sine, cosine, and tangent are not one-to-one, so their inverses would not be functions at all.

Related AP Precalculus Guides

• 3.3 Sine and Cosine Function Values
• 3.6 Sinusoidal Function Transformations
• 3.4 Sine and Cosine Function Graphs
• 3.8 The Tangent Function
• 3.11 The Secant, Cosecant, and Cotangent Functions
• 3.12 Equivalent Representations of Trigonometric Functions