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Angles and Degree Measure Chapter 4 Sec 1 Angles and Degree Measure

How do you describe angles and angular movement? Essential Question How do you describe angles and angular movement? Key Vocabulary: Initial side Terminal side Linear speed Angular speed

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 Angles that have a counterclockwise rotation have a positive measure. 90º or π/2 positive 0º or 2π 180º or π 270º or 3π/2

Clockwise means negative Angles that have a clockwise rotation have a negative measure. – 270º or –3π/2 –180º or –π 0º or –2π Negative –90º or – π/2

Coterminal Angles Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations. Since one rotation equals 2π, the measures of coterminal angles differ by multiples of 2π.

Radian Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in radians.

Since the circumference of a circle is 2πr units. Radian…still Since the circumference of a circle is 2πr units.

For the positive angle subtract 2π to obtain a coterminal angle. Example 1 For the positive angle subtract 2π to obtain a coterminal angle. For the negative angle add 2π to obtain a coterminal angle.

Complementary and Supplementary Angles Two positive angles α and β are complementary if their sum is π/2 or 90°. Two positive angles are supplementary if their sum is π or 180°. Find the complement and supplement of the following complement supplement complement supplement Because 5π/6 is greater than π/2 it has no complement. (Remember complements are positive angles.)

Find (if possible) the complement and supplement of the angle. Guided Practice Determine two coterminal angles in radian measure (one positive and one negative) for each angle. Find (if possible) the complement and supplement of the angle. complement supplement

A second way to measure angle is in the terms of degree, denoted by °. One degree = 1/360 To measure angles it is convenient to mark degrees on a circle. So a full revolution is 360°, a half is 180° a quarter is 90°… We see 360° is one revolution so 360°= 2π rad and 180° = π rad Thus

Degree/Radian Conversion 30° 45° 60° 90° 180° 360°

Convert from degrees to radians. Example 2 Convert from degrees to radians. 135o 540° – 270° Convert from radians to degrees.

Arc Length Arc Length For a circle of radius r, a central angle θ intercepts an arc of length s is given by s = r θ Length of circular arc where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measurement of θ equal the arc length A circle has a radius of 4 inches. Find the length of the arc intercepted by central angle of 240°.

Linear and Angular Speed 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, then the linear speed of the particle is: Linear speed = Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is Angular speed = Linear speed measures how fast the particle moves and angular speed measures how fast the angle changes.

In one revolution, the arc length traveled is Example 4 The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand. In one revolution, the arc length traveled is The time required for one revolution of the second hand is t = 1 minute or 60 seconds So the linear speed of the tip of the second hand is

A 15-inch diameter tire on a car makes 9.3 revolutions per second. Example 5 A 15-inch diameter tire on a car makes 9.3 revolutions per second. Find the angular speed of the tire in radians per second. Find the linear speed of the car. Because each revolution generates 2π radians, it follows that the tire turns (9.3)(2π)= 18.6π radians per second. So the angular speed is length traveled is The linear speed of the tire is

How do you describe angles and angular movement? Essential Question How do you describe angles and angular movement?

Chapter 4 Section 1 Text Book Pg 265 – 267 Show all work for credit. Daily Assignment Chapter 4 Section 1 Text Book Pg 265 – 267 #5 – 21 Mod4 #31 – 55 Mod4 #71 – 87 Mod4, #76 Show all work for credit.