Section 7 4 Evaluating and Graphing Sine and

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 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 Terminal Side Ѳ = 115 o Ѳ = 225 o Reference angle = 180 -115 = 65 o Reference angle = 225 -180 = 45 o

Example 12 150 o 2 Reference Angle 1 2 30 o 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 360. 1320 – 3(360) = 240 and 240 – 180 =60 180 o 1320 o or 240 o 60 o 1

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

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

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 45 -45 -90 right triangle the legs are the same length. Lets consider such a triangle whose legs have length 1. Then its hypotenuse has length c When we split a square In half diagonally we create a 45 -45 -90 Right triangle. 1 1

A 30 -60 -90 Right Triangle When we split an equilateral triangle in half we create 2 30 -60 -90 right triangles. 30 o 2 2 60 o 2 30 o 1 1 1

Example : In this triangle find sinѲ, cos Ѳ, and tan Ѳ 5 3 Ѳ 4

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

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. 1 0 o -1 90 o 180 o 360 o

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

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

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