| 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 b