9.2 Angles of Rotation Term Initial side Terminal side Positive s Negative s Coterminal s Angle of rotation Standard position Definition Where the rotation starts The ra on the positive - ais Also referred to as 0 Where the rotation stops Angles that go counterclockwise from 0 Angles that go clockwise from 0 terminal side -225 135 initial side Angles that share a terminal side, or stop at the same place. Formed b rotating the terminal side and keeping the initial side in place An is in standard position when its verte is at the origin and one ra is on the positive -ais. Starting at 0, and rotating in the positive direction, through which Quadrants will ou pass, and in what order?
Drawing Angles in Standard Position Draw and label the terminal sides of these rotation s. Then tell the Quadrant (or ais) in which the terminal side lies. Angle Quad. Angle Quad. A) 75 F) 315 B) 195 G) 390 C) -120 H) -45 D) -210 I) 630 E) 270 J) -240
Finding Coterminal Angles Which pairs of s above are coterminal? What is the degree measure that separates the s in each pair? From this, we can develop the following rule for determining what other rotation s are coterminal with an of degrees: If an measures, then it is coterminal with s measuring ( 360n). Finding Reference Angles Because rotation s can be ver large, the are often described using a reference, which is defined as the positive acute formed b the terminal side of θ and the -ais. For eample, a rotation of 135 has a 45 reference. Determine the reference for each rotation below. 45 Reference Rotation 135 A B C 320 207?? -150? D E F 508 53-280
Find measures for rotation s (between -360 and 360 ) that have the given characteristics. 1. A positive that terminates in Quadrant III with a 71 reference 2. A negative that terminates in Quadrant III with a 17 reference 3. A positive that terminates in Quadrant II with a 24 reference 4. A positive that terminates in Quadrant IV with a 24 reference Finding Values of Trigonometric Functions in Angles of Rotation In addition to finding values of trigonometric functions for s in right tris, we can also define the same functions in terms of s of rotation. Consider an in standard position, whose terminal side intersects a circle of radius r. We can think of the radius as the hpotenuse of a right tri: The point where the terminal side of the intersects the circle tells us the lengths of the two legs of the tri. Now, we can define the trigonometric functions in terms of, and :
We can also etend these functions to include non-acute s. Eample 1: The point (-3, 4) is a point on the terminal side of an in standard position. Determine the values of the si trigonometric functions of the. Solution: Notice that the is more than 90 degrees, and that the terminal side of the lies in the second quadrant. This will influence the signs of the trigonometric functions. Note: 1. The value of r depends on the coordinates of the given point. You can alwas find the value of r using the Pthagorean Theorem. 2. The values of the trigonometric functions ma be positive or negative depending upon the quadrant in which the terminal side of the lies. Therefore, it is important to include the signs of the - and -coordinates when determining the value of the trigonometric functions.
Practice Problems: 1. P( ) is a point on the terminal side of θ in standard position. Find the eact value of the si trigonometric functions for θ. sin θ= cos θ= tan θ= csc θ= sec θ= cot θ= 2. P( ) is a point on the terminal side of θ in standard position. Find the eact value of the si trigonometric functions for θ. sin θ= cos θ= tan θ= csc θ= sec θ= cot θ=