Section 7-4

Evaluating and Graphing Sine and Cosine

Copy and Complete the chart

Quadrant

Sign of:

x y Cos Θ Sin Θ

I

II

III

IV

Reference Angles

• Let α (Greek Alpha) be an acute angle in standard position. Suppose that α = 20°. Notice that the terminal ray of α = 20° and the terminal ray of 180° - α = 160° are symmetric in the y-axis. If the sine and cosine of α = 20° are known, then the sine and cosine of 160° can be deduced.

Reference Angles

Sin 160° = y = sin 20°Cos 160° = x = - cos 20°

122 yx

Reference Angle

• The angle α = 20° is called the reference angle for the 160° angle. It is also the reference angle for the 200° and 340° angles shown below.

Reference Angles

• In general, the acute angle α is the reference angle for the angles 180° - α, 180° + α, and 360° - α as well as all coterminal angles. In other words, the reference angle for any angle Θ is the acute positive angle formed by the terminal ray of Θ and the x-axis.

Reference Angles

• Is the angle between 0° and 360° (or 0 and 2π)?

• What quadrant is the angle in?

• Is sin or cos positive or negative in that quadrant?

• Apply the reference angle formula.

Θ180°- Θπ - Θ

Θ - 180° Θ - π

360°- Θ2π - Θ

Determine the Sign

** The sine or cosine of the reference angle for a given angle gives the absolute value of the sine or cosine of the given angle. The quadrant in which the given angle lies determines the sign of the sine or cosine of the given angle.

Using Calculators or Tables

• The easiest way to find the sine or cosine of most angles is to use a calculator that has the sine and cosine functions. Always be sure to check whether the calculator is in degree or radian mode.

• If you do not have access to a calculator, there are tables at the back of the book. Instructions on how to use the tables are on p. 800

Finding Sines and Cosines of Special Angles

• Because angles that are multiples of 30° and 45° occur often in mathematics, it can be useful to know their sine and cosine values without resorting to a calculator or table. To do this, you need to know the following facts:

Finding Sines and Cosines of Special Angles

In a 30°-60°-90° triangle, the sides are in the ratio (note that in this ratio, 1 corresponds to the side opposite the 30° angle, to the side opposite the 60° angle, and 2 to the side opposite the 90° angle.)

2:3:1

3

Finding Sines and Cosines of Special Angles

In a 45°-45°-90° triangle the sides are in the ratio , or .

2:1:12:2:2

• These facts are used in the diagrams below to obtain the values of sin Θ and cos Θ for Θ = 30°, Θ = 45°, and Θ = 60°.

Special Angles

• Although the table only gives the sine and cosine values of special angles from 0° to 90°, reference angles can be used to find other multiples of 30° and 45°.

• For example:

2245cos315cos

2130sin210sin

Special Angles

Special Angles

• As the table suggests, sin θ and cos θ are both one-to-one functions for the first quadrant θ

Graphs of Sine and Cosine• To graph the sine function, imagine a particle on the unit

circle that starts at (1, 0) and rotates counter clockwise around the origin. Every position (x, y) of the particle corresponds to an angle θ, where sin θ = y by definition. As the particle rotates through the four quadrants, we get the four pieces of the sine graph shown below.

• I From 0° to 90°, the y-coordinate increases from 0 to 1.• II From 90° to 180°, the y-coordinate decreases from 1 to

0.• III From 180° to 270°, the y-coordinate decreases from 0

to -1.• IV From 270° to 360°, the y-coordinate increases from -1

to 0.• Copy graph on p. 278

• Use the unit circle to complete the table of values.

• Then use the table of values to graph sine and cosine

The Sine Graph

The Sine Graph

The Sine Graph

• Since the sine function is periodic with a fundamental period of 360°, the graph above can be extended left and right as shown below.

The Cosine Graph

The Cosine Graph

The Cosine Graph

• To graph the cosine function, we analyze the x-coordinate of the rotating particle in a similar manner. The cosine graph is shown below.

Example

• Express each of the following in terms of a reference angle.

a.Sin 120°

b.Cos (-120°)

c.Sin 690°

Example

• Use a calculator or table to find the value of each expression to four decimal places.

a.Sin 43°

b.Cos 55°

c.Cos 230.46°

d.Sin 344.1°

Example

• Give the exact value of each expression in simplest radical form.

a.Sin 30° b. Sin

c. Cos (-120°) d. cos

Example

1. Find the exact value of each expression:

67sin

67cos 150sinlog4

Example

• Sketch the graph of y = - sin x and y = on the same set of axes. How many solutions does the equation –sin x = have?

3x

3x