Section 3.1 Radian Measure

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1 Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle that has its vertex at the center of a circle. The measure of the central angle whose intercepted arc is equal in length to the radius of the circle is equal to 1 radian. Definition If a central angle in a circle with radius r intercepts an arc on the circle of length s, then the measure of, in radians, is given by s in radians r Note: The units of s and r must be the same. English Math The measure of is 4 degrees. The measure of is 4 radians. Example 1 Find the measure (in radians) of a central angle,, that intercepts an arc on a circle of radius r with indicated arc length s. a) r = 10 cm, s = 2 cm b) r = 2 in, s = 4 in 1

2 Converting Between Degrees and Radians Recall that the circumference of a circle is 2 r. This is also the arc length of the circle, s. Let θ = 60º. Since the arc length is the circumference of the circle, then s = 2πr The central angle of the circle (in radians) will be: θ = So, 60º = radians. Then, 180º = radians. Therefore, To convert Degrees to Radians, multiply the degree measure by To convert Radians to Degrees, multiply the radian measure by. Example 2: Convert from degrees to radians. Leave answers in terms of. a) 0 b) 100 c) 120 Example : Convert from radians to degrees. a) 2 b) 5 12 c) 8 9 2

3 ?????? What is the difference between 0 and 0 radians??????? We can now draw the unit circle with special angles in both degrees and radians. Example 4: Evaluate cos 4 exactly. Evaluate 2 sin exactly.

4 Example 5: Find the reference angles for each angle given. a) 4 b) 11 6 c) 5 Example 6: Find the exact value of the following expressions. a) 7 sin 4 b) 5 tan 6 c) cot 2 d) csc 2 4

5 Section.2 Arc Length and Area of a Circular Sector Arc Length Recall from Section.1 that if a central angle in a circle with radius r intercepts an arc on the circle of length s, then the measure of, in radians, is given by s r. r We can use this formula to solve for the arc length, s. This formula is only true when is in radians. What if is in degrees? Example 1 Find the exact length of each arc made by the indicated central angle and radius of each circle. a), r 4mm b) 15, r 1800 km 5

6 Example 2 Find the exact length of each radius given the arc length and central angle of each circle. a) 5 s m, b) s m, 0 Example A low Earth orbit (LEO) satellite is in a approximate circular orbit 00 kilometers above the surface of the Earth. If the ground station tracks the satellite when it is within a 45 cone above the tracking antenna (directly overhead), how many kilometers does the satellite cover during the ground station track? Assume the radius of the earth is 6400 kilometers. Round to the nearest kilometer. 6

7 Area of a Circular Sector A Piece of the Pie English Math The ratio of the area of the sector to the area of the circle The ratio of the central angle, r to the angle of one full rotation These must be equal. Solve for A. This formula is only true when is in radians. What if is in degrees? Example 4 Find the area of the circular sector given the indicated radius and central angle. Round answers to three significant digits. a) 5, 1 mi 6 r b) 14, r.0 ft 7

8 Example 5 A windshield wiper that is 11 inches long (blade and arm) rotates 65. If the rubber part is 7 inches long, what is the area cleared by the wiper? Round to the nearest inch. 8

9 Section. Linear and Angular Speeds In this section, we will look at circular motion. Circular motion can be defined in terms of : s Linear speed (speed along the circumference of a circle), v, and t Angular speed (speed of angle rotation), where is given in radians. t The variable t is time. Example 1 Find the linear speed of a point that moves with constant speed in a circular motion if the point travels along the circle an arc length s = 12 ft in time t = min. Example 2 Find the distance traveled (arc length) of a point that moves with constant speed v along a circle in time t. v = 60 mi/hr, t = 15 min Example Find the angular speed (radians/second) associated with rotating a central angle in time t. 1 a), t sec b) 120, t 0 sec 4 6 9

10 Relationship Between Linear and Angular Speeds Recall the Ford Expedition problem. The upgrade would cause odometer and speedometer errors. This is because angular speed (rotations of tires per second), radius (of the tires), and linear speed (speed of the automobile) are all related. HOW????? If a point, P, moves at a constant speed along the circumference of a circle with radius r, then the linear speed, v, and the angular speed,, are related by: v vr or r Example 4 Find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius r and angular speed. 5 rad, r 9in sec 10

11 Example 5 Find the distance a point travels along a circle, s, over a time t given the angular speed and radius of the circle r. Round to three significant digits. rad r 2 mi, 6, t 11sec sec Example 6 The planet Jupiter rotates every 9.9 hours and has a diameter of 88,846 miles. If you re standing on its equator, how fast are you traveling (linear speed)? 11

12 Section.4 Definition of Trig Functions: The Unit Circle Approach Consider the unit circle: (x, y) x cos r y sin r So, the point (x, y) could be written as: Definition III If the point (x, y) is any point on the unit circle, and is the arc distance from the point (1, 0) to the point (x, y) along the circumference of the unit circle, then: 1 y cos x sec tan x x 1 x sin y csc cot y y Since is a real number, the trigonometric functions are often called circular functions. For each of the special angles, there is a corresponding point on the unit circle at which the terminal side of intersects. 12

13 Example 1: Find the exact values of a) 7 sin 4 b) 5 cos 6 c) tan 2 5 Example 2: Find all six trig functions for the angle. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = Example : Find all values of t (0 t < 2π), where sin t. 2 Even and Odd Functions: If f ( x) f ( x), then f( x ) is an EVEN function. If f ( x) f ( x), then f( x ) is an ODD function. 1

14 Consider the following: sin sin cos cos Cosine is an function because Sine is an function because Example 4: Use even and odd functions to find the exact values of the following: a) sin 4 b) cos 4 c) tan( 225 ) Example 5: Prove that Cosecant is an odd function What about Secant? Tangent? Cotangent? 14

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