flip the input and output of sine, cosine, and tangent. They're crucial for solving equations and finding angles when given trigonometric values. These functions have restricted domains and ranges to ensure unique solutions.
Exact values of inverse trig functions for common angles are essential to know. They help simplify expressions and solve problems without a calculator. Understanding these values and how to use technology for more complex calculations is key to mastering inverse trigonometry.
Inverse Trigonometric Functions
Inverse trigonometric functions
Inverses of the trigonometric functions sine, cosine, and tangent
Denoted as arcsin, arccos, and arctan or sin−1, cos−1, and tan−1
Reverse the input and output values of the original functions (sinθ=21, then arcsin21=θ)
and differ from the original trigonometric functions
arcsinx: Domain [−1,1], Range [−2π,2π]
arccosx: Domain [−1,1], Range [0,π]
arctanx: Domain (−∞,∞), Range (−2π,2π)
Solve equations by applying inverse trigonometric functions
Isolate the trigonometric function and apply its inverse to both sides (cosθ=23, then θ=arccos23)
Principal values of inverse trigonometric functions are unique angles within their restricted ranges
Exact values of inverse expressions
Common angles have exact values for their inverse trigonometric functions
arcsin21=6π, arccos21=3π, arctan1=4π
arcsin22=4π, arccos22=4π, arctan3=3π
arcsin23=3π, arccos23=6π, arctan31=6π
Simplify expressions by recognizing these exact values
arcsin(sin3π)=3π because sin3π=23 and arcsin23=3π
These values can be visualized on the unit circle
Technology for inverse trig evaluation
Calculators and software can evaluate inverse trigonometric functions for any input value
Find arccos(−0.8) using a calculator: arccos(−0.8)≈2.4980 radians or 143.13∘
Graph inverse trigonometric functions using technology to visualize their behavior
Plot y=arctanx using a graphing calculator or software to see its domain, range, and asymptotes
Composite functions with inverse trig
Trigonometric and inverse trigonometric functions cancel each other when composed
sin(arcsinx)=x for −1≤x≤1
cos(arccosx)=x for −1≤x≤1
tan(arctanx)=x for all real numbers x
Solve equations involving composite functions by working from the inside out
tan(arccosx)=1
arccosx=4π because tan4π=1
x=cos4π=22 because arccos22=4π
Periodic functions and inverse relations
Trigonometric functions are periodic, repeating their values at regular intervals
Inverse trigonometric functions are not periodic, as they are derived from inverse relations of the original functions
Radians are often used to express angles in inverse trigonometric functions, providing a more natural way to describe rotations
Key Terms to Review (11)
Arccosine: Arccosine, denoted as $\arccos(x)$ or $\cos^{-1}(x)$, is the inverse function of the cosine function. It returns the angle whose cosine is a given number within the interval [0, π].
Arcsine: Arcsine is the inverse function of the sine function, denoted as $\sin^{-1}(x)$ or $\arcsin(x)$. It returns the angle whose sine is a given number within the range of $[-1, 1]$.
Arctangent: Arctangent, denoted as $\arctan(x)$ or $\tan^{-1}(x)$, is the inverse function of the tangent function. It returns the angle whose tangent is a given number.
Domain: The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. It represents the range of values over which the function can be evaluated.
Inverse cosine function: The inverse cosine function, denoted as $\cos^{-1}(x)$ or $\arccos(x)$, returns the angle whose cosine is $x$. It is defined for $-1 \leq x \leq 1$ and returns values in the range $0 \leq y \leq \pi$.
Inverse sine function: The inverse sine function, denoted as $\sin^{-1}(x)$ or $\arcsin(x)$, is the inverse operation of the sine function. It returns the angle whose sine value is $x$ within the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Inverse tangent function: The inverse tangent function, denoted as $\text{tan}^{-1}(x)$ or $\arctan(x)$, returns the angle whose tangent is $x$. It is the inverse operation of the tangent function and has a range of $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.
Inverse trigonometric functions: Inverse trigonometric functions are the inverse operations of the basic trigonometric functions (sine, cosine, and tangent). They are used to find angles when given a ratio of sides in right triangles.
One-to-one function: A one-to-one function is a function where each input corresponds to exactly one unique output, and each output corresponds to exactly one unique input. In other words, no two different inputs produce the same output.
Range: The range of a function is the set of all possible output values it can produce. It is determined by evaluating the function over its domain.
Right triangle: A right triangle is a triangle in which one angle measures exactly 90 degrees. The side opposite this angle is called the hypotenuse, and the other two sides are known as the legs.