Section 7-4 Evaluating and Graphing Sine and Cosine Objectives: To use the reference angles, calculators and tables and special angles to find the values to find the values of Sine and Cosine and graph their functions

Reference Angles Objective: Find the reference angle of a rotation and use it to find trigonometric function values Definition The reference angle for a rotation is the acute angle formed by the terminal side and the x-axis Ѳ = 115 o Reference angle = = 65 o Terminal Side Ѳ = 225 o Reference angle = = 45 o

Example o 150 o Reference Angle Find the sine, cosine, and tangent of 1320 o We can subtract multiples of 360 o We do this by 1320 by 360 and taking the larger part. Thus we subtract 3 multiples of – 3(360) = 240 and 240 – 180 = o 60 o 1320 o or 240 o

Example 2 Express 695 o in terms of a reference angle 695 o o = 335 o The reference angle for 335 o I s 360 o o = 25 o And Since 695 o is in the fourth quarter then Sin 695 o = - Sin 25 o

(0,r) (-r,0) (0,-r) (r,0) 90 o 180 o 270 o Terminal Side on an Axis If the terminal side of an angle falls on an axis

Using Calculators or Tables The easiest way to find the sine or cosine of most angles is to use a scientific calculator. Always be sure to check whether the calc is in degree or radian mode. If you do not have a calculator you can use the table at the back of this book on page 800 Find the value of each expression to four decimal places. a. Sin 122 o b. Cos 237 o c. Cos 5 o d. Sin(-2)

Special Angles Objective: Find the length of sides in special triangles In a right triangle the legs are the same length. Lets consider such a triangle whose legs have length 1. Then its hypotenuse has length c 1 1 When we split a square In half diagonally we create a Right triangle.

A Right Triangle When we split an equilateral triangle in half we create right triangles o 60 o 30 o 1 2

Example : In this triangle find sinѲ, cos Ѳ, and tan Ѳ Ѳ

Example 2: In ΔABC, b =40cm and <A = 60 o. What is the length of side c. 60 c b B AC

Graph of the Sine Curve

The Cosine Function

Graphs of Sine and Cosine To Graph the Sine Function plot the position of the function using the radian or degree value. 0o 0o 90 o 180 o 360 o 1

Period and Amplitude of Trig Functions Amplitude is range of Maximum and Minimum points on a graph So for function y = A SinBx Amplitude = |A| period

Try These Amplitude = |A| = |3| = 3 Period A = 3 and B= 2

The Unit Circle

Homework (1-17) odd Pg.279 Day 2: 19,24,27,29,31,33 Pg.280

Graphs of the Six Trig Functions