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

23 2

23

Coterminal angles: same initial side and terminal side

Name a negative coterminal angle:

23

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

27.4

32.3

3.2

6.1

Degree Measure

So………

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

2. Radians → degrees: multiply radians by

1802360

deg1801

1801

rad

rad

180

180

Convert to Radians:

1. 320°

2. 45 °

3. -135 °

4. 270 °

5. 540 °

Convert to Radians:

45.4

56.3

3.2

2.1

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

1. 2. 60°

43

Sketch the angle

3. 613

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.

hypopp

sinhypadj

cosadjopp

tan

opphyp

cscadjhyp

secoppadj

cot

RECIPROCALS

Evaluating Trig Functions

• Use the triangle to find the exact values of the six trig functions of θ.

hypotenuse

θ

3

4

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

21

6sin30sin 0

23

3sin60sin 0

23

6cos30cos 0

21

3cos60cos 0

31

6tan30tan 0

14

tan45tan 0

33

tan60tan 0

Trig Identities

• Reciprocal Identities

csc1sin

sec1cos

cot1tan

sin1csc

cos1sec

tan1cot

Trig Identities (cont)

• Quotient Identities

cossintan

sincoscot

Evaluating Using the Calculator

• sin 63°

• tan (36°)

• sec (5°)

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?

• You are 200 yards from a river. Rather than walk directly to the river, you walk 400 yards along a straight path to the river’s edge. Find the acute angle θ between this path and the river’s edge.

• Find the length c of the skateboard ramp.