Jay Abramson

Key Concepts

8.1 Graphs of the Sine and Cosine Functions

Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of $2 π .$
The function $\sin x$ is odd, so its graph is symmetric about the origin. The function $\cos x$ is even, so its graph is symmetric about the y-axis.
The graph of a sinusoidal function has the same general shape as a sine or cosine function.
In the general formula for a sinusoidal function, the period is $P = \frac{2 π}{| B |} .$ See Example 1.
In the general formula for a sinusoidal function, $| A |$ represents amplitude. If $| A | > 1,$ the function is stretched, whereas if $| A | < 1,$ the function is compressed. See Example 2.
The value $\frac{C}{B}$ in the general formula for a sinusoidal function indicates the phase shift. See Example 3.
The value $D$ in the general formula for a sinusoidal function indicates the vertical shift from the midline. See Example 4.
Combinations of variations of sinusoidal functions can be detected from an equation. See Example 5.
The equation for a sinusoidal function can be determined from a graph. See Example 6 and Example 7.
A function can be graphed by identifying its amplitude and period. See Example 8 and Example 9.
A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See Example 10.
Sinusoidal functions can be used to solve real-world problems. See Example 11, Example 12, and Example 13.

8.2 Graphs of the Other Trigonometric Functions

The tangent function has period $π .$
$f (x) = A \tan (B x - C) + D$ is a tangent with vertical and/or horizontal stretch/compression and shift. See Example 1, Example 2, and Example 3.
The secant and cosecant are both periodic functions with a period of $2 π .$ $f (x) = A \sec (B x - C) + D$ gives a shifted, compressed, and/or stretched secant function graph. See Example 4 and Example 5.
$f (x) = A \csc (B x - C) + D$ gives a shifted, compressed, and/or stretched cosecant function graph. See Example 6 and Example 7.
The cotangent function has period $π$ and vertical asymptotes at $0, \pm π, \pm 2 π, ...$
The range of cotangent is $(- \infty, \infty),$ and the function is decreasing at each point in its range.
The cotangent is zero at $\pm \frac{π}{2}, \pm \frac{3 π}{2}, ...$
$f (x) = A \cot (B x - C) + D$ is a cotangent with vertical and/or horizontal stretch/compression and shift. See Example 8 and Example 9.
Real-world scenarios can be solved using graphs of trigonometric functions. See Example 10.

8.3 Inverse Trigonometric Functions

An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
For any trigonometric function $f (x),$ if $x = f^{- 1} (y),$ then $f (x) = y .$ However, $f (x) = y$ only implies $x = f^{- 1} (y)$ if $x$ is in the restricted domain of $f .$ See Example 1.
Special angles are the outputs of inverse trigonometric functions for special input values; for example, $\frac{π}{4} = \tan^{- 1} (1) and \frac{π}{6} = \sin^{- 1} (\frac{1}{2}) .$ See Example 2.
A calculator will return an angle within the restricted domain of the original trigonometric function. See Example 3.
Inverse functions allow us to find an angle when given two sides of a right triangle. See Example 4.
In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, $\sin (\cos^{- 1} (x)) = \sqrt{1 - x^{2}} .$ See Example 5.
If the inside function is a trigonometric function, then the only possible combinations are $\sin^{- 1} (\cos x) = \frac{π}{2} - x$ if $0 \leq x \leq π$ and $\cos^{- 1} (\sin x) = \frac{π}{2} - x$ if $- \frac{π}{2} \leq x \leq \frac{π}{2} .$ See Example 6 and Example 7.
When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See Example 8.
When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See Example 9.