1.1 Radian and Degree Measure

Introduction: Trigonometry – measurement of triangles

Applications of trigonometry: physics, engineering, surveying, and architecture

Angles: rotating a ray (half-line) about its endpoint

Represent angles with a , b , q

Positive angles – counterclockwise

Negative angles – clockwise

Radian Measure

Measure of an angle – amount of rotation from initial side to terminal side

Radian – measure of central angle q that subtends an arc s equal in length to the radius r of the circle

q = s/r

full circle = 2p ~ 6.28 radians

half revolution = p

quarter revoltion = p /2

sixth revolution = p /3

p /2, -3p /2

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p /2 < q < p | 0 < q < p /2

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Q2 | Q1

-p , p ------------------------------------------------- 0, 2p

Q3 | Q4

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p < q < 3p /2 | 3p /2 < q < 2p

3p /2, -p /2

Example 1: Sketch a) q = 13p /6, b) q = 3p /4, c) q = -2p /3

acute == 0 < q < p /2

obtuse == p /2 < q < p

a , b complementary iff a + b = p /2

a , b supplementary iff a + b = p

Example 2: Find the complements and supplements of

a) 2p /5 b) 4p /5

Degree Measure

1° (one degree) = 1/360 of a complete revolution

360° = 2p 180° = p rad

1. to convert ° to rad, multiply ° by p rad / 180°
2. to convert rad to ° , multiply rad by 180° / p rad

Example 3: Convert a) 135° , b) 540° , c) -270°

Example 4: Convert a) -p /2, b) 9p /2, c) 2 rad

Degrees, Minutes and Seconds

1’ = 1/60°

1" = 1/60’ = 1/3600°

Example 5: Convert 152° 15’29" to °

Applications:

Example 6: A circle has a radius of 4". Find the length of the arc cut off by 240° .

Solution: 240° = 240 * p /180 = 4p /3 rad

s = rq = 4(4p /3) = 16p /3 ~ 16.76 rad

Speed = Distance / Time = s / t

Angular Speed = q / t

Example 7: The second hand of a clock is 4". Find the speed of the tip of the second hand.

Solution: t = 60 sec. = 1 min.

s = 2p (4) = 8p inches

speed = s / t = 8p / 60 sec ~ 0.419 in / sec

Example 8: A lawn roller 30" in diameter makes 1.2 revolutions/sec,

1. find the angular speed:

Solution: angular speed = q / t = 2.4 p rad / 1 sec

= 2.4 p rad / sec

2. find the speed:

Solution: speed = (1.2 rev / sec) (30p in / rev)