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Chapter 13 Sec 2. Angles and Degree Measure. Standard Position. An angle in standard position has its vertex at the origin and initial side on the positive x– axis . terminal side. initial side. Positively Counterclockwise. - PowerPoint PPT Presentation
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Chapter 13 Sec 2
Angles and Degree Measure
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Algebra 2 Chapter 13 Sections 2 & 3
• An angle in standard position has its vertex at the origin and initial side on the positive x–axis.
initial side
terminal side
Standard Position
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Algebra 2 Chapter 13 Sections 2 & 3
• Angles that have a counterclockwise rotation have a positive measure.
130
0º
90º
180º
270º
Positively Counterclockwise
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Algebra 2 Chapter 13 Sections 2 & 3
• Angles that have a clockwise rotation have a negative measure.
– 130
0º
– 90º
– 180º
– 270º
Clockwise means negative
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Algebra 2 Chapter 13 Sections 2 & 3
Now let’s look at angle measures 30, 150, 210, and 330.
180°(1, 0)(–1, 0)
30º30º30º30º
30°150°
210°330°
They all form a 30° angle with the x-axis, so they should all have the same sine, cosine, and tangent values…only the signs will change!
The angle to the nearest x-axis is called the reference angle.
All angles with the same reference angle will have the same trig values except for sign changes.
Unit Circle
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Algebra 2 Chapter 13 Sections 2 & 3
Unit Circle• A unit circle is a circle with radius 1. • If we have an angle between 0o and 90o
in standard position. Let P(x, y) be the point of intersection. If a perpendicular segment is drawn we create a right triangle, where y is opposite θ and x is adjacent to θ.
• Right triangles can be formed for angles greater than 90o, simply use the reference angle.
yyhypopp
1
sin xxhypadj
1
cos
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Algebra 2 Chapter 13 Sections 2 & 3
Radian…still• A point P(x, y) is on the unit circle if and only
if its distance from the origin is 1.
• The radian measure of an angle is the length of the corresponding arc on the unit circle.
• Since
P(x, y)s
α
radians 2360 thus,1 and 2 rrC
radians. 2
90 and radians 081 o... S
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Algebra 2 Chapter 13 Sections 2 & 3
Degree/Radian Conversion
degree180
Radians radians
180
Degree
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Algebra 2 Chapter 13 Sections 2 & 3
Example 1a. Change 115o to radian measure in terms of π..
b. Change radian to degree measure.
180115 115 oo
3623
87
180
87
87
5.157
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Algebra 2 Chapter 13 Sections 2 & 3
30° and 45° Radians• You will need to know these conversions.
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Algebra 2 Chapter 13 Sections 2 & 3
• Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.
• Since one rotation equals 360, the measures of coterminal angles differ by multiples of 360.
300 – 360 =
300 60
60 + 360 = – 60420
Coterminal Angles
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Algebra 2 Chapter 13 Sections 2 & 3
Example 2Find one positive and one negative coterminal angle.
a. 45o
45o + 360o = 405o and 45o – 360 o = –315o
b. 225o
225o + 360o = 585o and 225o – 360 o = –135o
Chapter 13 Sec 3
Trigonometric Functions
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Algebra 2 Chapter 13 Sections 2 & 3
Radius other than 1.• Suppose we have a hypotenuse with a length
other than 1. For our example we’ll use r as the length.
• In standard position r extends from the Origin to point P(x, y).
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Algebra 2 Chapter 13 Sections 2 & 3
Quadrantal Angle• If a terminal side of an angle coincides with one
of the axes, the angle is called a quadrantal angle. See below for examples:
• A full rotation around the circle is 360o. Measures more than 360o represent multiple rotations.
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Algebra 2 Chapter 13 Sections 2 & 3
To find the values of trig functions of angles greater than 90, you will need to know how to find the measures of the reference angle. If θ in nonquadrantal, its reference angle is formed by the terminal side of the given angle and the x-axis.
Reference Angles
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Algebra 2 Chapter 13 Sections 2 & 3
Example 1Find the reference angle for each angle.
a. 312o Since 312o is between 270o and 360o the terminal
side is in fourth quad. Therefore, 360o – 312o = 48o.
b. –195o
the coterminal angle is 360o – 195o = 165o this put us in the second quadrant so… 180o – 165o = 15o
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Algebra 2 Chapter 13 Sections 2 & 3
0°/360°(1, 0)
90° (0, 1)
270° (0, –1)
180°(–1, 0)
Students
Sine values are positive
(csc, too)
All
All values are positive
Take
Tangent values are positive
(cot, too)
Calculus
Cosine values are positive
(sec, too)
Determining Sign
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Algebra 2 Chapter 13 Sections 2 & 3
Example 2Find the values of the six trigonometric functions for angle θ in standard position if a point with coordinates (–15, 20) lies on the terminal side.
256252015 22 r
54
2520sin
53
2515cos
34
1520tan
45
2025csc 3
515
25sec
43
2015cot
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Algebra 2 Chapter 13 Sections 2 & 3
Example 3Find the values of the six trigonometric functions Suppose θ is an angle in standard position whose terminal side lies in the Quadrant III. If find the remaining five trigonometric functions of θ.
7
means III Quad 7
7
342
222
222
y
y
y
y
yxr
47sin
34sec
43cos
37
37tan
773
73cot
774
74csc
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Algebra 2 Chapter 13 Sections 2 & 3
Daily Assignment• Chapter 13 Sections 2 & 3• Study Guide
• Pg 177• #4 – 7
• Pg 178 – 180 Odd