Chapter 4 Trigonometric Functions. Angles Trigonometry means measurement of triangles. In...
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Transcript of Chapter 4 Trigonometric Functions. Angles Trigonometry means measurement of triangles. In...
![Page 1: Chapter 4 Trigonometric Functions. Angles Trigonometry means measurement of triangles. In Trigonometry, an angle often represents a rotation about a point.](https://reader035.fdocument.org/reader035/viewer/2022062304/56649f515503460f94c73b5e/html5/thumbnails/1.jpg)
Chapter 4
Trigonometric Functions
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AnglesTrigonometry means measurement of triangles.
In Trigonometry, an angle often represents a rotation about a point. Thus, the angle θ shown is the result of rotating its initial ray to its terminal ray.
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Standard Position of an AngleAn angle whose vertex is the origin and whose initial side coincides with the positive x-axis is an angles in standard position.
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Positive and Negative AnglesPositive angles are generated by counterclockwise rotation.
Negative angles are generated by clockwise rotation.
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Labeling angles
Angles are labeled with Greek letters such as alpha ( ) and beta ( ), and theta ( ), as well as uppercase letters.
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Quadrantal Angles
If the terminal ray of an angle in standard position lies in the first quadrant, the angle is said to be a first-quadrantal angle. The second-, third- and fourth-quadrant angles are similarly defined.
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What is a radian?A common unit for measuring smaller
angles is the degree, of which there are 360° in one revolution.
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Illustration of Arc LengthWhen an arc of a circle has the same length as the radius of the circle, the measure of the central angle, is by definition one radian.
One radian is approximately
57.2958 degrees.
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Section 4.1, Figure 4.6,Illustration of Six Radian
LengthsThere are 360° or 2 radians in one revolution.
360° = 2 radians180° = radians90° = /2 radians45° = /4 radians
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Common Radian Angles
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Conversion Formulas1. To convert degrees to radians,
multiply degrees by
2. To convert radians to degrees, mutliply radians by
180
radians
radians180
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“Leave in terms of Pi”
Convert 250° to radians
and radians to degrees.4
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List the Special Angles
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Coterminal AnglesTwo angles are coterminal if they have the same initial and terminal rays.
“Different nameFor the same thing”
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Finding Coterminal AnglesYou can find an angle coterminal to a given angle by adding or subtracting ????
For the positive angle find a
Positive coterminal angle.
Negative coterminal angle.
6
13
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Finding Coterminal Angles
For the negative angle , find positive and negative coterminal angles.
Find two angles, one positive and one negative, that are coterminal with the angle .
3
2
4
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Complementary and Supplementary Angles
Two positive angles are complementary (complements of each other) if their sum is 90° or
radians.
Two positive angles are supplementary (supplements of each other) if their sum is 180° or
radians.
2
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Arc length
Arc length is measured in radians by the formula:
srWhere S is arc length
r is the radius is the central angle
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A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°.
A circle has a radius of 9 feet. Find the length of the arc intercepted by a central angle of /4.
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Linear and Angular SpeedImagine a particle moving at a constant speed along
the edge of a circle. The distance the particle travels with respect to time is the particle’s linear speed.
Linear speed =
Linear speed measures how fast the particle moves.
ArcLength
Times
t
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Linear and Angular SpeedImagine a spinning circle. Consider a central angle of
that circle.
Angular speed measures how fast the angle changes per a unit of time.
Angular speed =
Angular speed measures how fast the angle changes.
CentralAngle
Time
t
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See examples on page 254 for Linear and Angular speed.
Goes back to this idea
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Classwork…
Page 255Know vocab 1-105 - 59 odd (skip 21, 29)77 - 87 odd96 - 98