08-31 Lecture: Inverse Trigonometric Functions and Domain Restrictions
The lecture provides a comprehensive review of inverse trigonometric functions, focusing on why their domains must be restricted and how these restrictions define the functions' ranges. The instructor explains that standard trigonometric functions like sine, cosine, and tangent are not one-to-one and fail the horizontal line test, thus preventing inversion. To create their inverses (arcsine, arccosine, and arctangent), a specific "chunk" of the original function's graph is selected that is one-to-one and covers the entire range of values. The lecture details the conventionally agreed-upon domain restrictions for sine ([-π/2, π/2]), cosine ([0, π]), and tangent ((-π/2, π/2)), and explains how these restrictions determine the ranges and corresponding quadrants for the output angles of their inverse functions. The concept of concavity is also introduced as a visual differentiator between the original and inverse graphs. The instructor emphasizes that understanding these ranges is crucial for solving problems, which will be the focus of the next lecture on September 1, 2026.
Invertibility and the One-to-One Condition
A function can only be inverted if it is one-to-one, meaning it passes the horizontal line test.
Functions like y = x² are not one-to-one because elements in the range correspond to more than one domain value.
Similarly, trigonometric functions like sine and cosine are periodic and fail the horizontal line test "horribly," as the test line would pierce the graph an infinite number of times.
Restricting the Domain to Enable Inversion
To invert a function that is not one-to-one, we must restrict its domain to a "chunk" of the graph that is one-to-one.
There are two main goals when selecting this chunk:
It must be one-to-one.
It must capture every value in the original function's range (e.g., from -1 to 1 for sine).
Domain Restriction for Sine
To create a one-to-one function, the domain of the sine function is conventionally restricted to the interval [-π/2, π/2].
This specific interval is chosen because it is one-to-one and covers the full range of sine values from -1 to 1.
Properties of Arcsine (sin⁻¹(x))
The process of finding an inverse involves swapping the x and y variables. This means the domain and range of the original function are interchanged.
Domain: [-1, 1] (These are the input ratios).
Range: [-π/2, π/2] (These are the output angles).
The output of an arcsine function is always an angle, which will be in either Quadrant 1 or Quadrant 4 (specifically, the negative angle values).
Graph of Arcsine
The graph of arcsine is a reflection of the restricted sine graph over the line y=x.
Key points on the graph are (1, π/2), (0, 0), and (-1, -π/2).
Unlike the sine wave, the arcsine graph is not periodic.
The concavity also swaps. The restricted sine graph is concave up then concave down, while the arcsine graph is concave down then concave up.
Domain Restriction for Cosine
The interval [-π/2, π/2] cannot be used for cosine because it would not be one-to-one and would only yield positive values.
The conventionally agreed-upon domain restriction for the cosine function is the interval [0, π].
This chunk is one-to-one and captures the full range of cosine values from -1 to 1.
Properties of Arccosine (cos⁻¹(x))
Domain: [-1, 1].
Range: [0, π].
The output of an arccosine function is always an angle in Quadrant 1 (where cosine is positive) or Quadrant 2 (where cosine is negative).
Graph of Arccosine
The graph is generated by swapping the coordinates of points on the restricted cosine graph.
Key points include (1, 0), (0, π/2), and (-1, π).
The graph is visually distinct from the original cosine segment.
Domain Restriction for Tangent
For tangent, one full wave from -π/2 to π/2 is selected. This chunk is one-to-one and covers the entire range of values from negative infinity to positive infinity.
The domain is restricted to the open interval (-π/2, π/2), excluding the endpoints where vertical asymptotes occur.
Properties of Arctangent (tan⁻¹(x))
Domain: All real numbers ((-∞, ∞)).
Range: (-π/2, π/2).
The output of an arctangent function is an angle in Quadrant 1 or Quadrant 4.
Graph of Arctangent
The graph of arctangent is the restricted tangent graph laid on its side.
The vertical asymptotes of the tangent function become horizontal asymptotes for the arctangent function at y = -π/2 and y = π/2.
Calculator Use
Calculators are programmed with these same range restrictions. When asked for an inverse trigonometric value, a calculator will only provide an output within the defined range.
For example, arcsin(-1) will yield -π/2, not a coterminal angle like 3π/2.
Problem Solving
Understanding the specific range for each inverse trig function is crucial for solving equations. An answer like 7π/4 for an arcsine problem would be marked wrong because it is outside the range [-π/2, π/2], even though it's in the correct quadrant. The correct answer would be -π/4.
The focus of the lecture on September 1, 2026, will be on practicing problems using these range rules.
1. For the upcoming bell ringer on September 1, 2026, be prepared to graph the tangent function.
2. Get out your Chromebook, open Desmos in testing mode (with the green banner), and ensure you know how to use it. If you don't have a Chromebook, get one from the library.
3. Work on the bell ringer sheet provided in class. Desmos may be helpful for the last three problems but not the first two.