4.1: radian and degree measure objectives: to use radian measure of an angle to convert angle...
TRANSCRIPT
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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
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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)
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Positive Angles
Negative Angle
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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
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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
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• 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
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(1 revolution)
½ a revolution =
¼ a revolution
1/6 a revolution=
1/8 a revolution=
3602
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Quadrant 1Quadrant 2
Quadrant 3 Quadrant 4
20
2
2
3 2
2
3
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Coterminal angles: same initial side and terminal side
Name a negative coterminal angle:
2
3
2
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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
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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
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Convert to Radians:
1. 320°
2. 45 °
3. -135 °
4. 270 °
5. 540 °
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Convert to Radians:
4
5.4
5
6.3
3.2
2.1
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Sketching Angles in Standard Position: Vertex is at origin, start at 0°
1. 2. 60°
4
3
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Sketch the angle
3. 6
13
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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
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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”
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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
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Warm-Up
• Evaluating Trig Functions– Use the triangle to find the exact values of the six
trig functions of θ.
13
θ
5
12
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Evaluating Trig Functions
• sinθ = 7/15– Use the given information to find the exact values
of the other 5 trig functions of θ.
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Special Right Triangles
45-45-90 30-60-90
45°
45°
1
1
2
30°
60°
21
3
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Evaluating Trig Functions for 45°
• Find the exact value of sin 45°, cos 45°, and tan 45°
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Evaluating Trig Functions for 30° and 60°
• Find the exact values of sin60°, cos 60°, sin 30°, cos 30°
30°
60°
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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
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Trig Identities
• Reciprocal Identities
csc
1sin
sec
1cos
cot
1tan
sin
1csc
cos
1sec
tan
1cot
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Trig Identities (cont)
• Quotient Identities
cos
sintan
sin
coscot
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Evaluating Using the Calculator(Pay attention to units and mode)• sin 63°
• sec 36°
• tan (π/2)
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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
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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?
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• Find the length c of the skateboard ramp.