| Quadrant I: [latex]\theta = \widetilde{\theta}[/latex] | [latex]\hphantom{0000}[/latex] | Quadrant II: [latex]\theta = 180^{o} - \widetilde{\theta}[/latex] |
| Quadrant III: [latex]\theta = 180^{o} + \widetilde{\theta}[/latex] | [latex]\hphantom{0000}[/latex] | Quadrant IV: [latex]\theta = 360^{o} - \widetilde{\theta}[/latex] |
Chapter 4: Trig Functions
Chapter 4 Summary and Review
Key Concepts
- We can use angles to describe rotation. Positive angles indicate rotation in the counterclockwise direction; negative angles describe clockwise rotation.
- We define the trigonometric ratios of any angle by placing the angle in standard position and choosing a point on the terminal side, with [latex]r = \sqrt{x^2 + y^2}{.}[/latex]
The Trigonometric Ratios.
If [latex]\theta[/latex] is an angle in standard position, and [latex](x,y)[/latex] is a point on its terminal side, with [latex]r = \sqrt{x^2 + y^2}{,}[/latex] then
[latex]\sin \theta = \dfrac{y}{r}~~~~~~~~~ \cos \theta = \dfrac{x}{r}~~~~~~~~~ \tan \theta = \dfrac{y}{x}[/latex]
- To construct a reference triangle for an angle:
- Choose a point [latex]P[/latex] on the terminal side.
- Draw a line from point [latex]P[/latex] perpendicular to the [latex]x[/latex]-axis.
- The reference angle for [latex]\theta[/latex] is the positive acute angle formed between the terminal side of [latex]\theta[/latex] and the [latex]x[/latex]-axis.
- The trigonometric ratios of any angle are equal to the ratios of its reference angle, except for sign. The sign of the ratio is determined by the quadrant.
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To find an angle [latex]\theta[/latex] with a given reference angle [latex]\widetilde{\theta}{:}[/latex] - There are always two angles between [latex]0°[/latex] and [latex]360°[/latex] (except for the quadrantal angles) with a given trigonometric ratio.
- Coterminal angles have equal trigonometric ratios.
- To solve an equation of the form [latex]\sin \theta = k{,}[/latex] or [latex]\cos \theta = k{,}[/latex] or [latex]\tan \theta = k{,}[/latex] we can use the appropriate inverse trig key on a calculator to find one solution (or a coterminal angle). We use reference angles to find a second solution between [latex]0°[/latex] and [latex]360°{.}[/latex]
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Angles in a Unit Circle.
Let [latex]P[/latex] be a point on a unit circle determined by the terminal side of an angle [latex]\theta[/latex] in standard position. Then the coordinates [latex](x,y)[/latex] of [latex]P[/latex] are given by
[latex]x = \cos \theta,~~~~~~y = \sin \theta[/latex]
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Coordinates.
If point [latex]P[/latex] is located at a distance [latex]r[/latex] from the origin in the direction specified by angle [latex]\theta[/latex] in standard position, then the coordinates of [latex]P[/latex] are
[latex]x = r \cos \theta ~~~~ {and} ~~~~ y = r \sin \theta[/latex]
- Navigational directions for ships and planes are sometimes given as bearings, which are angles measured clockwise from north.
- Periodic functions are used to model phenomena that exhibit cyclical behavior.
- The trigonometric ratios [latex]\sin \theta[/latex] and [latex]\cos \theta[/latex] are functions of the angle [latex]\theta{.}[/latex]
- The period of the sine function is [latex]360°{.}[/latex] Its midline is the horizontal line [latex]y = 0{,}[/latex] and the amplitude of the sine function is 1.
- The graph of the cosine function has the same period, midline, and amplitude as the graph of the sine function. However, the locations of the intercepts and of the maximum and minimum values are different.
- We use the notation [latex]y = f(x)[/latex] to indicate that [latex]y[/latex] is a function of [latex]x[/latex]; that is, [latex]x[/latex] is the input variable and [latex]y[/latex] is the output variable.
- The tangent function has period [latex]180°{.}[/latex] It is undefined at odd multiples of [latex]90°[/latex] and is increasing on each interval of its domain.
- The angle of inclination of a line is the angle [latex]\alpha[/latex] measured in the positive direction from the positive [latex]x[/latex]-axis to the line. If the slope of the line is [latex]m{,}[/latex] then[latex]\tan \alpha = m[/latex] where [latex]0° \le \alpha \le 180°{.}[/latex]
Amplitude, Period, and Midline.
- The graph of
[latex]y = A\cos\theta ~~{or}~~ y = A\sin\theta[/latex]
has amplitude [latex]\lvert{A}\rvert{.}[/latex] - The graph of
[latex]y =\cos B\theta ~~{or}~~ y = \sin B\theta[/latex]
has period [latex]\dfrac{360°}{\lvert{B}\rvert}{.}[/latex] - The graph of
[latex]y = k + \cos\theta ~~{or}~~ y =k + \sin\theta[/latex]
has midline [latex]y = k{.}[/latex] - The graph of [latex]y = k + A\sin B\theta[/latex] has amplitude [latex]A{,}[/latex] period [latex]\dfrac{360°}{B}{,}[/latex] and midline [latex]y = k{.}[/latex] The same is true for the graph of [latex]y = k + A\cos B\theta{.}[/latex]
- The graph of
- Functions that have graphs shaped like sines or cosines are called sinusoidal.
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Periodic Function.
The function [latex]y = f(x)[/latex] is periodic if there is a smallest value of [latex]p[/latex] such that
[latex]f(x + p) = f(x)[/latex]
for all [latex]x{.}[/latex] The constant [latex]p[/latex] is called the period of the function.