13-2 Angles and the Unit Circle Hubarth Algebra II

An angle is in standard position when the vertex is at the origin and one ray is on the positive x-axis. The ray on the x-axis is the initial side of the angle; the other ray is the terminal side of the angle Standard Position y Initial side x Terminal side Ex. 1 Measuring an Angle in Standard Position Find the measure of the angle. The angle measures 60° more than a right angle of 90°. Since 90 + 60 = 150, the measure of the angle is 150°.

The measure of an angle is positive when it rotates counterclockwise from the initial side to The terminal side. The measure is negative when the rotation is clockwise. Ex. 2 Sketching an Angle in Standard Position Sketch each angle in standard position. a. 48° b. 310° c. –170° Two angles are coterminal angles if they have the same terminal side y 135 Angles that have measures 135 and -225 are coterminal. x -225

Ex. 3 Real-World Connection The Aztec calendar stone has 20 divisions for the 20 days in each month of the Aztec year. An angle on the Aztec calendar shows the passage of 16 days. Find the measures of the two coterminal angles that coincide with the angle. The terminal side of the angle is of a full rotation from the initial side. 16 20 • 360° = 288° 16 20 To find a coterminal angle, subtract one full rotation. 288° – 360° = –72° Two coterminal angle measures for an angle on the Aztec calendar that show the passage of 16 days are 288° and –72°.

The unit circle has a radius of I unit and its center at the origin of the coordinate plane. Points on the unit circle are related o the periodic functions.

Ex. 4 Finding the Cosine and Sine of an Angle Find the cosine and sine of 135°. From the figure, the x-coordinate of point A  is – , so cos 135° = – , or about –0.71. 2 Use a 45°-45°-90° triangle to find sin 135°. opposite leg = adjacent leg 0.71   Simplify. =    Substitute. 2 The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 135° –0.71 and sin 135° 0.71.

Ex. 5 Finding Exact Values of Cosine and Sine Find the exact values of cos (–150°) and sin (–150°). Step 1:  Sketch an angle of –150° in standard position. Sketch a unit circle. x-coordinate = cos (–150°) y-coordinate = sin (–150°) Step 2:  Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.) The triangle contains angles of 30°, 60°, and 90°.

Step 3: Find the length of each side of the triangle. hypotenuse = 1 The hypotenuse is a radius of the unit circle. shorter leg = The shorter leg is half the hypotenuse. 1 2 1 2 3 longer leg = 3 = The longer leg is 3 times the shorter leg. 3 2 1 Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so cos (–150°) = – , and sin (–150°) = – .

Practice One full rotation contains 360 degrees. How many degrees are in one quarter of a rotation? In one half of a rotation? In three quarters of a rotation? 90°, 180°, 270° 2. Sketch each angle in standard position. a. 85° b. −320° c. 180° Do on the board 3. Find another angle coterminal with 198° by adding one full rotation. 558°