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TRIGONOMETRY CHAPTER 3: RADIAN MEASURE AND THE CIRCULAR FUNCTIONS 3.1 RADIAN MEASURE RADIAN: o The figure below shows an angle in standard position along with a circle of radius r. The vertex of is at the center of the circle. Angle intercepts an arc on the circle equal in length to the radius of the circle. Therefore, angle is said to have a measure of 1 radian. r r r CONVERTING BETWEEN DEGREES AND RADIANS o The circumference of a circle (the distance around a circle) is given by the formula C 2 r , where r is the radius of the circle. This tells us that there are 2 radians around a circle. Consider an angle of 360°. This is a complete circle, so 360° intercepts an arc on the circle equal in length to 2 times the radius of the circle. Therefore, we have 360 2 radians. This gives us the following relationship: 180 1 1 360 2 radians radians 2 2 1 So we have, 180 radians 1 180 radian 1 radian 180 Example: Convert 45° to radians 45 45 radian 180 4 radian 5 radian to degrees 6 5 5 180 radian 6 6 150 Example: Convert Example: Find cot cot 3 4 3 3 cot 180 4 4 cot 135 1 2 EQUIVALENT ANGLE MEASURES IN DEGREES AND RADIANS 0° or 0 radians 30° or radians 6 45° or radians 4 60° or radians 3 90° or radians 2 2 120° or radians 3 3 135° or radians 4 5 150° or radians 6 180° or radians 7 210° or radians 6 5 225°or radians 4 4 240° or radians 3 3 270° or radians 2 5 300° or radians 3 7 315° or radians 4 11 330° or radians 6 360° or 2 radians sin 0 cos 1 tan 0 1 2 3 2 3 3 cot Undefined 3 2 2 2 2 1 1 3 2 1 2 1 0 Undefined 3 2 12 3 2 2 1 2 3 2 -1 3 3 csc Undefined 2 2 3 3 2 2 3 3 2 2 3 3 0 Undefined 1 -2 2 3 3 3 2 2 sec 1 3 3 -1 2 2 3 2 33 2 0 12 -1 23 0 3 3 Undefined 3 -1 2 33 Undefined -2 2 2 1 1 2 2 3 2 3 3 -2 2 33 0 Undefined -1 2 2 33 2 2 -1 2 2 12 3 0 Undefined 3 2 1 2 3 2 2 2 2 -1 12 3 2 0 1 3 3 0 3 3 -1 3 2 3 3 -2 Undefined 1 Undefined 3 3.2 APPLICATIONS OF RADIAN MEASURE ARC LENGTH OF A CIRCLE: The length of an arc is proportional to the measure of its central angle. T Q s r R radians 1 radian O r P r s r 1 o Arc Length: The length s of the arc intercepted on a circle of radius r by a central angle of measure radians is given by the product of the radius and the radian measure of the angle, or s r , in radians. Example: Find the radius of a circle in which a central angle of 2 radians intercepts an arc of length 3 feet. s s r r 3 feet 2 o Area of a Sector: The area of a sector of a circle of radius r and central angle is given by r A 1 2 r , in radians. 2 4 Example: Find the area of a sector of a circle having a radius of 18.3 m and a central angle of 125°. First you have to change the angle to radian measure. So we 25 radians . Now we can solve the problem have 125 180 36 using the area for a sector of a circle. 1 2 r 2 1 25 18.32 2 36 365 m 2 A 3.3 CIRCULAR FUNCTIONS OF REAL NUMBERS THE CIRCULAR FUNCTIONS o The radius (r) is always 1. Therefore, from our earlier equation (derived from the distance formula) we have 1 x 2 y 2 . o Trigonometric functions for the circular functions Recall that that the radian measure of is related to the arc length s. Here, r = 1, so s = r gives us s = . y y x tan s , x 0 sin s y cos s x x 1 1 1 x 1 csc s , y 0 cot s , y 0 sec s , x 0 y y x (0, 1) (-1, 0) (1, 0) (0, -1) 5 Since sin s = y and cos s = x, we can replace x and y in the equation x 2 y 2 1 and obtain the identity cos 2 s sin 2 s 1 . o Domain and Range of the Circular Functions Since the ordered pair (x, y) represents a point on the unit circle, 1 x 1 and 1 y 1, so 1 cos s 1 and 1 cos s 1. The domain of cos s and sin s is the set of all real numbers, since cos s and sin s exist for any value of s The domains of tan s and sec s are restricted since x has a value of 0 at plus any multiple of , positive 2 or negative. So we write the domains for tan s and sec s as follows: s | s n , n . 2 The domains of cot s and csc s are restricted since y has a value of 0 at any multiple of , positive or negative. So we write the domains for cot s and csc s as follows: s | s n , n . The Unit Circle: The figure below shows the relationship between degree and radian measure. This is something you need to memorize! It will make the rest of the course much easier. 6 3.4 LINEAR AND ANGULAR VELOCITY Linear Velocity o In many situations we need to know how fast a point on a circle is moving or how fast the central angle is changing. Suppose that point P moves at a constant speed along a circle of radius r and center O. The measure of how fast the position of P is changing is called linear velocity. If v represents velocity, then we have distance time s v t velocity where s is the length of the arc traced by point P at time t. This formula is just d rt with s as the distance, v as the rate, and t is still time. P s O r B Angular Velocity o Consider the figure above. As point P moves along the circle, OP rotates around the origin. Since OP is the terminal side of POB , the measure of the angle changes as P moves along the 7 circle. The measure of how fast POB is changing is called angular velocity. Angular velocity is denoted by is given as , in radians, t where is the measure of POB at time t. is expressed as radians per unit of time. Recall from section 3.2 that the arc length of a circle was measured by s r , in radians . Now that we know the meaning of , we have an alternate formula for linear velocity. v s r r r . t t t This formula relates linear and angular velocities. Note that a radian is a “pure number”, with no units associated with it. This is why you get some length per unit of time with no extra unit like degrees. o Example: Radius of a gear. A gear is driven by a chain that travels 1.46 m/s. Find the radius of the gear if it makes 46 rev/minute. Solution: v = 1.46 m/s. One complete revolution is 2 radians, 46 2 23 46 and 46 seconds is minute, so 60 60 15 radians per second. Now we have only one unknown in the formula for linear velocity v r . Substituting for v and we have 1.46 23 v r 1.46 r r 23 0.303 meters. 15 15 8