4.1: radian and degree measure objectives: to use radian measure of an angle to convert angle...

4.1: Radian and Degree Measure

Objectives:•To use radian measure of an angle•To convert angle measures back and forth between radians and degrees•To find coterminal angle

We are going to look at angles on the coordinate plane… An angle is determined by rotating a ray about its

endpoint Starting position: Initial side (does not move) Ending position: Terminal side (side that rotates) Standard Position: vertex at the origin; initial side

coincides with the positive x-axis Positive Angle: rotates counterclockwise (CCW) Negative Angle: rotates clockwise (CW)

Positive Angles

Negative Angle

1 complete rotation: 360⁰Angles are labeled with Greek letters: α (alpha), β (beta), and θ (theta)Angles that have the same initial and terminal

sides are called coterminal angles

RADIAN MEASURE (just another unit of measure!)

Two ways to measure an angle: radians and degrees For radians, use the central angle of a circle

s=rr

• s= arc length intercepted by angle• One radian is the measure of a

central angle, Ѳ, that intercepts an arc, s, equal to the length of the radius, r

• One complete rotation of a circle = 360°• Circumference of a circle: 2 r• The arc of a full circle = circumference

s= 2 rSince s= r, one full rotation in radians= 2 =360 °

, so just over 6 radians in a circle

28.62

(1 revolution)

½ a revolution =

¼ a revolution

1/6 a revolution=

1/8 a revolution=

3602

Quadrant 1Quadrant 2

Quadrant 3 Quadrant 4

20

2

2

3 2

2

3

Coterminal angles: same initial side and terminal side

Name a negative coterminal angle:

2

3

2

You can find an angle that is coterminal to a given angle by adding or subtracting

Find a positive and negative coterminal angle:

2

2

7.4

3

2.3

3.2

6.1

Degree Measure

So………

Converting between degrees and radians:1. Degrees →radians: multiply degrees by

2. Radians → degrees: multiply radians by

180

2360

deg180

1

1801

rad

rad

180

180

Convert to Radians:

1. 320°

2. 45 °

3. -135 °

4. 270 °

5. 540 °

Convert to Radians:

4

5.4

5

6.3

3.2

2.1

Sketching Angles in Standard Position: Vertex is at origin, start at 0°

1. 2. 60°

4

3

Sketch the angle

3. 6

13

4.3 Right Triangle Trigonometry

Objectives:• Evaluate trigonometric functions of acute

angles• Evaluate trig functions with a calculator• Use trig functions to model and solve real

life problems

Right Triangle Trigonometry

hypotenuse

θ

Side adjacent to θ

Side opposite θ

Using the lengths of these 3 sides, we form six ratios that define the six trigonometric functions of the acute angle θ.

sine cosecantcosine secanttangent cotangent

*notice each pair has a “co”

Trigonometric Functions

• Let θ be an acute angle of a right triangle.

hyp

oppsin

hyp

adjcos

adj

opptan

opp

hypcsc

adj

hypsec

opp

adjcot

RECIPROCALS

Warm-Up

• Evaluating Trig Functions– Use the triangle to find the exact values of the six

trig functions of θ.

13

θ

5

12

Evaluating Trig Functions

• sinθ = 7/15– Use the given information to find the exact values

of the other 5 trig functions of θ.

Special Right Triangles

45-45-90 30-60-90

45°

45°

1

1

2

30°

60°

21

3

Evaluating Trig Functions for 45°

• Find the exact value of sin 45°, cos 45°, and tan 45°

Evaluating Trig Functions for 30° and 60°

• Find the exact values of sin60°, cos 60°, sin 30°, cos 30°

30°

60°

Sine, Cosine, and Tangent of Special Angles

2

1

6sin30sin 0

2

3

3sin60sin 0

2

3

6cos30cos 0

2

1

3cos60cos 0

3

1

6tan30tan 0

14

tan45tan 0

33

tan60tan 0

Trig Identities

• Reciprocal Identities

csc

1sin

sec

1cos

cot

1tan

sin

1csc

cos

1sec

tan

1cot

Trig Identities (cont)

• Quotient Identities

cos

sintan

sin

coscot

Evaluating Using the Calculator(Pay attention to units and mode)• sin 63°

• sec 36°

• tan (π/2)

Applications of Right Triangle Trigonometry

• Angle of elevation: the angle from the horizontal upward to the object

• Angle of depression: the angle from the horizontal downward to the object

Word Problems

• A surveyor is standing 50 feet from the base of a large tree. The surveyor measure the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

• Find the length c of the skateboard ramp.