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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.