Glossary

8.3 Inverse Trigonometric Functions

2 min read•june 24, 2024

Inverse trigonometric functions flip the script on regular trig functions. They find angles when you know the sine, cosine, or tangent value. These functions have unique domains and ranges, making them useful for solving tricky angle problems.

Mastering inverse trig functions opens up new ways to tackle complex math challenges. From finding exact values to using technology, these tools are key for advanced problem-solving in trigonometry and beyond.

Inverse Trigonometric Functions

Inverse trigonometric functions

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  • Inverses of the standard trigonometric functions undo the original function
  • Denoted with a superscript -1: sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, tan⁡−1\tan^{-1}
  • Domain and range differ from the original functions
    • Inverse sine (sin⁡−1\sin^{-1} or arcsin) domain: [-1, 1], range: [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]
    • Inverse cosine (cos⁡−1\cos^{-1} or arccos) domain: [-1, 1], range: [0, π\pi]
    • Inverse tangent (tan⁡−1\tan^{-1} or arctan) domain: (−∞-\infty, ∞\infty), range: (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2})
  • Apply inverse functions to find the angle given the trigonometric value
    • sin⁡θ=x\sin \theta = x, then θ=sin⁡−1x\theta = \sin^{-1} x
    • cos⁡θ=x\cos \theta = x, then θ=cos⁡−1x\theta = \cos^{-1} x
    • tan⁡θ=x\tan \theta = x, then θ=tan⁡−1x\theta = \tan^{-1} x
  • Inverse functions are the result of reflecting the original function over the line y = x

Exact values of inverse trigonometry

  • Find exact values using the and common angle measures (π6\frac{\pi}{6}, π4\frac{\pi}{4}, π3\frac{\pi}{3})
  • Examples:
    • sin⁡−1(12)=π6\sin^{-1}(\frac{1}{2}) = \frac{\pi}{6}
    • cos⁡−1(−22)=3π4\cos^{-1}(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4}
    • tan⁡−1(1)=π4\tan^{-1}(1) = \frac{\pi}{4}
  • finds the angle measure given the trigonometric value
  • is often used when working with inverse trigonometric functions

Technology in inverse trigonometry

  • Scientific calculators have buttons for inverse trigonometric functions
    • Labeled as "", "", "" or "arcsin", "arccos", "arctan"
  • Enter the trigonometric value and press the corresponding button
    • To find sin⁡−1(0.8)\sin^{-1}(0.8), enter 0.8 and press "sin^-1" or "arcsin"
  • Calculator results in radians unless set to degree mode

Composite inverse trigonometric functions

  • Composite functions apply one function to the result of another (sin⁡(cos⁡−1x)\sin(\cos^{-1} x))
  • Solve by working from the innermost function outward
    1. Evaluate the inverse trigonometric function first
    2. Apply the outer trigonometric function to the result
  • Simplify the expression if possible
    • sin⁡(cos⁡−1(32))\sin(\cos^{-1} (\frac{\sqrt{3}}{2}))
      1. cos⁡−1(32)=π6\cos^{-1} (\frac{\sqrt{3}}{2}) = \frac{\pi}{6}
      2. sin⁡(π6)=12\sin(\frac{\pi}{6}) = \frac{1}{2}
    • Final answer: 12\frac{1}{2}
  • is a key concept in understanding these types of problems

Trigonometric Ratios and the Unit Circle

  • (sine, cosine, tangent) are derived from the unit circle
  • The unit circle is a circle with a radius of 1 centered at the origin
  • Points on the unit circle correspond to angle measures and their trigonometric values
  • Understanding the unit circle helps in visualizing inverse trigonometric functions

Key Terms to Review (31)

Angle of Depression: The angle of depression is the acute angle formed between the horizontal line of sight and the downward line of sight to an object that is below the observer's eye level. It is a concept used in trigonometry and navigation to determine the vertical distance or elevation of an object relative to the observer's position.
Angle of Elevation: The angle of elevation is the angle between the horizontal line of sight and the line of sight to an object above the observer. It is a crucial concept in right triangle trigonometry and the application of inverse trigonometric functions.
Arccosine: Arccosine, also known as the inverse cosine function, is a trigonometric function that allows us to find the angle whose cosine is a given value. It is an essential concept in understanding right triangle trigonometry, inverse trigonometric functions, and the law of cosines for non-right triangles.
Arcsine: The arcsine, also known as the inverse sine function, is a trigonometric function that represents the angle whose sine is a given value. It is used to find the angle when the ratio of the opposite side to the hypotenuse of a right triangle is known.
Arctangent: The arctangent is the inverse trigonometric function that gives the angle whose tangent is a given value. It is used to find the angle in a right triangle given the ratio of the opposite and adjacent sides.
Cos^-1: The inverse cosine function, denoted as cos^-1 or arccos, is a trigonometric function that gives the angle whose cosine is a given value. It is used to find the angle when the cosine of that angle is known.
Cramer’s Rule: Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, utilizing determinants. It provides an explicit formula for the solution of the system.
Even function: An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.
Even Function: An even function is a mathematical function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Horizontal asymptote: A horizontal asymptote is a horizontal line that a graph approaches as the input values go to positive or negative infinity. It indicates the end behavior of a function's output.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function's graph approaches as the input value (x) approaches positive or negative infinity. It represents the limit of the function as it gets closer and closer to this line, without ever touching it.
Inverse Composition Identity: The inverse composition identity is a fundamental concept in mathematics that relates the composition of functions to their inverses. It states that the composition of a function with its inverse function results in the identity function, which maps each element to itself.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Inverse Sine Identity: The inverse sine identity, also known as the arcsine identity, is a fundamental relationship in trigonometry that connects the inverse sine function to the original sine function. It allows for the simplification and evaluation of expressions involving the inverse sine function.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, odd functions exhibit symmetry about the origin.
Odd Function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that the graph of an odd function is symmetric about the origin, with the graph being a reflection across both the $x$-axis and the $y$-axis.
Principal Value: The principal value refers to the primary or main output of an inverse trigonometric function, representing an angle that corresponds to a given trigonometric ratio. In the context of inverse trigonometric functions, principal values are defined within specific intervals to ensure that each value is unique and corresponds to only one angle, thus avoiding ambiguity and maintaining consistency across mathematical calculations.
Properties of determinants: The determinant is a scalar value that is computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. It is crucial for solving systems of linear equations using Cramer's Rule.
Radian Measure: Radian measure is a way of expressing angles in terms of the ratio of the length of the arc subtended by the angle to the radius of the circle. It is a fundamental concept in trigonometry that provides a more natural and versatile way of working with angles compared to the more familiar degree measure.
Reciprocal Identity: The reciprocal identity is a fundamental concept in trigonometry that states the reciprocal relationship between certain trigonometric functions. It establishes a direct connection between the values of these functions, allowing for the conversion and simplification of trigonometric expressions.
Restricted Domain: The restricted domain of a function refers to the limited range of input values for which the function is defined. It represents the subset of the domain where the function can be evaluated without resulting in undefined or invalid outputs.
Sin^-1: The inverse sine function, denoted as sin^-1 or arcsin, is a trigonometric function that allows us to find the angle whose sine is a given value. It is the inverse operation of the sine function, which means it undoes the effect of the sine function.
Square matrix: A square matrix is a matrix with an equal number of rows and columns. It is often used in solving systems of linear equations and performing matrix operations.
System of three equations in three variables: A system of three equations in three variables consists of three linear equations, each containing three different variables. The solution is a set of values for the variables that satisfy all three equations simultaneously.
Tan^-1: The inverse tangent function, denoted as tan^-1 or arctan, is a trigonometric function that calculates the angle whose tangent is a given value. It is used to find the angle when the tangent ratio is known, reversing the process of the tangent function.
Trigonometric Ratios: Trigonometric ratios are the fundamental mathematical relationships between the sides and angles of a right triangle. These ratios, including sine, cosine, and tangent, are essential for understanding and applying right triangle trigonometry, the behavior of other trigonometric functions, inverse trigonometric functions, and solving trigonometric equations.
Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) of the coordinate plane. It is a fundamental concept in trigonometry, as it provides a visual representation of the relationships between the trigonometric functions and the angles they represent.
Vertical asymptote: A vertical asymptote is a line $x = a$ where a rational function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. It represents values that $x$ cannot take, causing the function to become unbounded.
Vertical Asymptote: A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the vertical limit of the function's behavior, indicating where the function's value becomes arbitrarily large or small.
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