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4.1 Radian & Degree Measure
Objectives: Describe angles. Use radian measure. Use degree measure and convert
between degree and radian measure. Use angles to model and solve real-life
problems.
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What is Trigonometry? Started as the measurement of
triangles. Applications: astronomy,
navigation, surveying. Developed into functions.
Applications: sound waves, light rays, planetary orbits, vibrating strings, pendulums, orbits of atomic particles.
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What is an Angle? An angle is formed by rotating a ray
about its endpoint.
Initial Side – starting position of the ray.
Terminal Side – position after rotation.
Vertex – endpoint of the ray.
Usually labeled with Greek letters.
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Angles in Standard Position Initial Side – lies on the
positive x-axis Vertex – located at the
origin. Positive Angle –
counter-clockwise rotation (towards positive y-axis)
Negative Angle – clock-wise rotation (towards negative y-axis)
Measure of an Angle The measure of an angle is the
amount of rotation from the initial side to the terminal side. Angles are measured in radians or degrees.
Radians One radian is the
measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.
Arc Length s = rθ, where θ is measured in radians
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Radians and Degrees Circum. of a circle =
2πr One revolution = 360° Therefore, 2π = 360°
Degrees
Radians
360° 2π180° π90° π/245° π/430° π/660° π/3
270° 3π/2
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Co-terminal Angles Have the same initial and terminal
sides.
To find co-terminal angles, add or subtract multiples of 2π (or 360°).
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Example 1 Find a positive and negative co-
terminal angle for each and then sketch the angles.
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2.3
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3.2
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13.1
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Complementary & Supplementary Complementary Angles
Supplementary Angles
Note: Must be positive angles.
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Angle Conversions Degrees to Radians
Radians = Degrees ·
Radians to Degrees Degrees = Radians ·
Note: π radians = 180°
180
180
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Example 5 A circle has a radius of 4 inches.
Find the length of the arc intercepted by a central angle of 240°.
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Linear Speed Consider a particle moving at a constant speed along a circular arc of radius r. Let s be the length of the arc traveled in time t. The linear speed of the particle is given by
That is, how fast is the particle moving along the arc?
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Angular Speed Consider the same particle moving at a constant speed along the same circular arc of radius r. If θ is the angle (in radians) corresponding to the arc length s, then the angular speed of the particle is given by
That is, how fast does the central angle change as the particle moves along the arc?
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Example 6 The second hand of a clock is 10.2
cm long. Find the linear speed of the tip of this second hand.
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Example 7 A lawn roller with a 10-
inch radius makes 1.2 revolutions per second.
a. Find the angular speed of the roller in radians per second.
b. Find the speed of the tractor that is pulling the roller.