Drill

Calculate:

1.sin 4.cot4 4

2.tan 5.sec6 6

3.cos3

Section 4.3

Students will calculate trig ratios using

reference triangles and the unit circle.

What is an angle?

• In trig, we think of an angle as a ray that has been rotated.

• The initial side is rotated about the vertex until the desired angle is formed.

Initial side

Rotates to new position

What is an angle?

• The initial side is the beginning position of the ray.

• The vertex is the endpoint of the ray.• The terminal side is the ending position

of the ray.

Initial sideTe

rmin

al sid

e

vertex

Location

• We will place the initial side of the angle on the positive x axis.

• We may rotate in the negative direction or the positive direction.

Ө

Initial side

Positive Angles

• Positive angle: formed by a counterclockwise rotation

+Ө

Negative angles

• Negative angle: formed by a clockwise rotation

-Ө

Which quadrant is it in?

• 285°• 250 °• -285• -110•

•

23

116

Coterminal Angles

• Have the same initial side and terminal side.

• Example:• 90˚, 450˚, -270˚ are all coterminal

• To find coterminal angles, add or subtract integer multiples of 360 ˚ (2 radians).

Find three coterminal angles

• 275˚• 35˚• 135˚

• radians

• radians

34

56

Evaluating trig functions on the coordinate plane: Q1

• Let θ be the acute angle in standard position whose terminal side contains the point (5, 3). Find the six trigonometric functions of θ.

Evaluating trig functions on the coordinate plane: Q2

• Let θ be any angle in standard position whose terminal side contains the point (-5, 3). Find the six trigonometric functions of θ.

One more example

• Let θ be any angle in standard position whose terminal side contains the point (-4, -3). Find the six trigonometric functions of θ.

Trigonometric Functions of Any Angle

• Let θ be any angle in standard position and let P(x, y) be any point on he terminal side of the angle (except the origin).

2 2r= x +ysin

cos

tan , 0

y

rx

ryx

x

csc , 0

sec , 0

cot , 0

ry

y

rx

xxy

y

Evaluating any angle

1. Draw a picture2. Find the reference angle

a. The reference triangle is the triangle that is formed by the horizontal axis and the terminal side.

3. Calculate the trig ratio(s).

Finding a reference angle

Quadrant IV360-Given angle

Quadrant IIIGiven angle - 180

Quadrant II180-Given angle

Quadrant IGiven angle isreference angle

Find the reference angle

1. 300°2. 210°3. 45°4. -330°

Example

• Evaluate the trig functions of 315˚.

Evaluate

• sin -210˚

• tan

• sec

23

34

The Unit Circle

• The unit circle is a circle of radius 1 centered at the origin.

• What is the coordinate of the intersection between the unit circle and the terminal side of the angle?

Using the unit circle

• Use the unit circle to easily calculate trig ratios of angles that are multiples of 30, 60, 45, or 90 degrees.

• Get out your “snowman” circle.

Why?

• Let t be an angle with vertex at the origin. • Then

• sin t = y• cos t = x• tan t = y/x• csc t = 1/y• sec t = 1/x• cot t = x/y

Evaluate

• sin (120˚)• tan ( )• csc ( )• sec ( )

3

415

623

Quadrantal Angle

• A quadrantal angle is an angle whose terminal side lies on an axis.

Find each of the following.

• sin (-90)• tan (180)• cos (- )• sec ( )• cot ( )

23

213

3

Homework

• Read 4.3, do #s 1 – 47 odd