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• Q: What is the greatest angle measure you recall seeing in Geometry? • Q: What do you think a negative angle measure might represent? Angles and Angle Measure 12.2 Standard position • Here the angle is positive since we see a counterclockwise rotation. • Initial side lies on pos. x-axis. • Vertex centered at origin. • Terminal side (blue) here is in Quad III. • Red arc represents angle rotation. Example 1 • Draw an angle with the given measure in standard position. Then tell in which quadrant the terminal side lies. • (a) 215o (b) 400o (c) -90o d) -335o • What would be a positive angle measure with the same standard position as -90o ? • How do you arrive at that measure? Coterminal Angle: • when two angles in standard position have terminal sides that coincide. **Find coterminal angles by adding or subtracting multiples of 360o from the original angle. Example 2 • Find one positive and one negative angle that are coterminal with: • a) -100o b) 575o Radians You can also measure angles in radians. To define a radian, consider a circle with radius r centered at the origin. Radian: the measure of an angle in standard position whose terminal side intercepts an arc of length r. • Because the circumference of a circle is 2πr, there are 2π radians in a full circle. • 360 = 2π radians • 180 = π radians Example 3 Draw the angle in standard position. Then determine which quadrant the angle lies in. a) 2𝜋 7 b) 11𝜋 6 c) 13𝜋 4 Example 3 Draw the angle in standard position. Then determine which quadrant the angle lies in. d) 3𝜋 − 7 e) 11𝜋 − 8 Coterminal Angles in radians: **Find coterminal angles by adding or subtracting multiples of 2π from the original angle. Example 4 Find one positive and one negative coterminal angle. a) 2𝜋 7 b) 7𝜋 − 6 Conversions Example 5: Convert. a) 320 to radians b) 5 12 to degree. Example 5: Convert. c) -245 to radians d) 3 4 to degree. Arc Length - Central Angle – an angle with a vertex at the center of the circle. Example 6 Find the length of each arc. Keep answer in terms of π. Example 7 • The steering wheel on a monster truck has a radius of 11 inches. How far does a point on the steering wheel travel if the wheel makes four fifths of a rotation?