Section 7-3 The Sine and Cosine Functions

Objective: To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

Review topics

Draw a circle: Label each quadrants as 1-4 Note the positive or negative x and y

values in each quadrant

Trig or Trick!

Draw an outline of your non-dominate hand on your paper. Spread your fingers so that your thumb and pinky make approximately 90 degrees.

http://mathrescue.blogspot.com/2012/08/trigonometry-evaluating-base-angles.html

Trig Trick!

Draw a UCIdentify the quadrants; what is the sign of (x,y) in each quadrant?

Q IQ II

Q III Q IV

(+,+)(-,+)

(-,-) (+,-)

(0,0)

(0,+1)

(-1,0)

(0,-1)

Review of geometry concepts

Given a right triangle, do you remember the definition of sin and cos ?

An angle in the standard position

We can study the same angle…

It is in the standard position.

Its vertex is on point (0,0).

It is on the coordinate system.

Now what if the same triangle was within a circle on

Sine and Cosine Functions

– We define the sine of θ, denoted sin θ, by:

sin y

rWe define the cosine of θ, denoted cos θ, by:

cos x

r

What if r=1

If r=1 what does sin θ and cos θ equal to?

sin θ=y cos θ=x

The Unit Circle

The circle of radius 1 is called the unit circle.

We can determine the value of many angles on the unit circle. Find A.) sin 90° B.) sin 450° C.) cos (-π)

Example 1 If the terminal ray of an angle θ in

standard position passes through (-3, 2), find sin θ and cos θ.

Solution:

On a grid, locate (-3,2)

Use this point to draw a right triangle, where one side is on the x-axis, and the hypotenuse is line segment between (-3,2) and (0,0).

Click here to see graph

Graph for last example

Example 2 Given θ is a 4th quadrant angle

sin 5

13

Find cos.

Click here to see graph

Graph of last example.

Unit Circle

The circle x2 + y2 = 1 has radius 1 and is therefore called the unit circle. This circle is the easiest one with which to work because, as the diagram shows, sin θ and cos θ are simply the y- and x-coordinates of the point where the terminal ray of θ intersects the circle.

Sin θ = y / r = y / 1 = y Cos θ = x / r = x / 1 = x When a circle is used to define the

trigonometric functions, they are sometimes called circular functions.

Example 4

Solve sin θ = 1 for θ in degrees.

Repeating Sin and Cos Values

Sin (θ + 360°) = sin θ Cos (θ + 360°) = cos θ Sin (θ + 2π) = sin θ Cos (θ + 2π) = cos θ Sin (θ + 360°n) = sin θ Cos (θ + 360°n) = cos θ Sin (θ + 2πn) = sin θ Cos (θ + 2πn) = cos θ

Homework sec 7.3 written exercises Day 1: Problems # 1-16 All Day 2 #17-28 , 33-42 ALL