Transcript Angles
4.1 Radian and Degree Measure Trigonometry- from the Greek “measurement of triangles” Deals with relationships among sides and angles of triangles and is used in astronomy, navigation, and surveying. Angles • An angle is two rays with the same initial point. • The measure of an angle is the amount of rotation required to rotate one side, called the initial side, to the other side, called the terminal side. • The shared initial point of the two rays is called the vertex of the angle. Angles in standard position • Standard position - the vertex is at the origin of the rectangular coordinate system and the initial side lies along the positive x-axis. • If the rotation of the angle is in the counterclockwise direction, then the angle is said to be positive. If the rotation is clockwise, then the angle is negative. Coterminal Angles • Two angles in standard position that have the same terminal side are said to be coterminal. You find coterminal angles by adding or subtracting 2 Radians vs. Degrees One radian is the central angle required to stretch the radius around the outside of the circle. Since the circumference of a circle is C 2r, it takes 2 radians to get completely around the circle once. Therefore, it takes radians to get halfway around the circle. 2 0, 2 3 2 Common Radian Angles, pg. 285 Angles between 0 and π/2 are acute Angles between π/2 and π are obtuse θ = π/2 Quad II π/2 < θ < π Quad I 0 < θ < π/2 θ=π Quad III Quad IV π < θ < 3π/2 3π/2 < θ < 2π θ = 3π/2 θ = 0, 2π Types of angles Complementary Angles : Angles that add up to 90 or 2 ά β Supplementary Angles: Angles that add up to 180or ά β One degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex. 1 180 radians 1radian 180 To convert degrees to radians, multiply the degrees by 180 To convert radians to degrees multiply the radians by 180 Convert the following degree measures to radian measure. a) 120° b) −315° c) 12° Convert the following radian measures to degrees. a) 5π/6 b) 7 c) 7 Arc Length: For a circle of radius r, a central angle θ intercepts an arc of length s given by s=rθ, where θ is measured in radians Example: Find the length of the arc that subtends a central angle with measure 120° in a circle with radius 5 inches Angular and Linear Velocity Angular Velocity (ω) is the speed at which something rotates. Therefore, which t means the rotation per unit time (how fast something is going around a circle). Linear Velocity (v) is the speed at which the outside tip of the radius is traveling. Therefore, v = rω. This equation considers the number of radii (since is expressed in radians) that travel around the circle during the rotation process. Example 1: A lawn roller with a 10-inch radius makes 1.2 revolutions per second. a.) Find the angular speed of the roller in radians per second. b.) Find the speed of the tractor that is pulling the roller in mi/hr. Example 2: The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of this hand. Example 3: An automobile is traveling at 65 mph. If each tire has a radius of 15 inches, at what rate are the tires spinning in revolutions per minute (rpm)? Assignment: • Page 291 # 5-21 odd, 35-65 odd, 75-85 odd, 95, 97