How many degrees are in each fraction of one complete revolution?
How many degrees are in each fraction of one complete revolution?
5. [latex]\dfrac{1}{4}[/latex]
6. [latex]\dfrac{1}{5}[/latex]
7. [latex]\dfrac{1}{6}[/latex]
8. [latex]\dfrac{1}{8}[/latex]
Algebra Refresher Answers
[latex]24[/latex]
[latex]24[/latex]
[latex]24[/latex]
[latex]24[/latex]
[latex]90°[/latex]
[latex]72°[/latex]
[latex]60°[/latex]
[latex]45°[/latex]
Learning Objectives
Use the coordinate definition of the trig ratios.
Find the trig ratios of supplementary angles.
Know the trig ratios of the special angles in the second quadrant.
Find two solutions of the equation [latex]\sin \theta = k[/latex].
Find the area of a triangle.
The town of Avery lies [latex]48[/latex] miles due east of Baker, and Clio is [latex]34[/latex] miles from Baker, in the direction [latex]35^{\circ}[/latex] west of north. How far is it from Avery to Clio?
We know how to solve right triangles using the trigonometric ratios. But the triangle formed by the three towns is not a right triangle, because it includes an obtuse angle of [latex]125^{\circ}[/latex] at [latex]B[/latex], as shown in the figure.
A triangle that is not a right triangle is called an oblique triangle. In this chapter we learn how to solve oblique triangles using the laws of sines and cosines. But first we must be able to find the sine, cosine, and tangent ratios for obtuse angles.
Angles in Standard Position
To extend our definition of the trigonometric ratios to obtuse angles, we use a Cartesian coordinate system. We put an angle [latex]\theta[/latex] in standard position as follows:
Place the vertex at the origin with the initial side on the positive [latex]x[/latex]-axis;
the terminal side opens in the counterclockwise direction.
We choose a point [latex]P[/latex] on the terminal side of the angle and form a right triangle by drawing a vertical line from [latex]P[/latex] to the [latex]x[/latex]-axis.
The length of the side adjacent to [latex]\theta[/latex] is the [latex]x[/latex]-coordinate of point [latex]P[/latex], and the length of the side opposite is the [latex]y[/latex]-coordinate of [latex]P[/latex]. The length of the hypotenuse is the distance from the origin to [latex]P[/latex], which we call [latex]r[/latex]. With this notation, our definitions of the trigonometric ratios are as follows.
Coordinate Definitions of the Trigonometric Ratios.
[latex]\cos \theta = \frac{x}{r}[/latex]
[latex]\sin \theta = \frac{y}{r}[/latex]
[latex]\tan \theta = \frac{y}{x}[/latex]
It doesn’t matter which point [latex]P[/latex] on the terminal side we use to calculate the trig ratios. If we choose some other point, say [latex]P'[/latex] with coordinates [latex]x', y'[/latex], as shown at right, we will get the same values for the [latex]\sin, \cos,[/latex] and tangent of [latex]\theta[/latex]. The new triangle formed is similar to the first one, so the ratios of the sides of the new triangle are equal to the corresponding ratios in the first triangle.
Example 3.1.
Find the values of [latex]\cos \theta[/latex