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Trigonometry
• (from Greek) the measurement of triangles
• Originally used in the development of astronomy, navigation and surveying
• After development of calculus and physical science in 17th century, used for rotations and vibrations, sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.
Definition of Radian:
One radian (rad) is the measure of a central angle that intercepts an arc s in equal length to the radius of the circle.
Radian Measure
r
s
rs 2
radians 24
2revolution
4
1
radians 2
2revolution
2
1
radians 36
2revolution
6
1
radians 612
2revolution
12
1
radians 2revolution 1
Complementary and Supplementary Angles
7
5 (b)
8
3 (a)
of supplement theand complement thefind possible, If
Conversion
.180
rad by degreesmultiply :radians convert to To
.rad
180by radiansmultiply :degrees convert to To
Arc Length
• A circle has a radius of 27 inches. Find the length of the arc intercepted by a central angle of 160o
Linear and Angular Speed(Constant speed along a circular path.)
• Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, the linear speed of the particle is
t
s
time
length arcSpeedLinear
• If is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is
t
time
angle centralSpeedAngular
Example:
• The second hand of a clock is 8 cm long. Find the linear speed of the tip of this second hand as it passes around the clock face.