Chapter 4 4-1 Radian and degree measurement Objectives O Describe Angles O Use radian measure O Use degree measure and convert between and radian measure Angles O As derived form the Greek language O Trigonometry means “measurement of triangles “ O Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy , navigation and surveying O Today, the use has expanded to involve rotations, orbits, waves, vibrations, etc. Angles O An angle is determined by rotating a ray (half-line) about its endpoint. Definitions O The initial side of an angle is the starting position of the rotated ray in the formation of an angle. O The terminal side of an angle is the position of the ray after the rotation when an angle is formed. O The vertex of an angle is the endpoint of the ray used in the formation of an angle. Standard Position O An angle is in standard position when the angle’s vertex is at the origin of a coordinate system and its initial side coincides with the positive x-axis. Positive and negative angles O A positive angle is generated by a counterclockwise rotation; whereas a negative angle is generated by a clockwise rotation. Coterminal O If two angles are coterminal, then they have the same initial side and the same terminal side. Radian Measure O The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. O One way to measure angles is in radians. O To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle. Radian Measure O Arc length = radius when 𝜃= 1 radian Radian How many radians are in a circle? O *In general, the radian measure of a central angle θ with radius r and arc length s is O θ = s/r. O We know that the circumference of a circle is 2πr. If we consider the arc s as being the circumference, we get θ = 2π r/r =2𝜋 Radians O This means that the circle itself contains an angle of rotation of 2π radians. Since 2π is approximately 6.28, this matches what we found above. There are a little more than 6 radians in a circle. (2π to be exact.) Therefore: A circle contains 2π radians. A semi-circle contains π radians of rotation. A quarter of a circle (which is a right angle) contains 𝜋2 radians of rotation Radians Coterminal angles O Two angles are coterminal when they have the same initial and terminal sides. O For instance, the angle 0 and 2𝜋 are 𝜋 coterminal, as are the angles 𝑎𝑛𝑑 13𝜋/6. 6 O You can find an angle that is coterminal to a given angle by adding or subtracting 2𝜋. Example#1 O For the positive angle 13𝜋 / 6, subtract 2𝜋 to obtain a coterminal angle. Example#2 O For the positive angle 3𝜋 / 4, subtract 2𝜋 to obtain a coterminal angle. Example#3 O For the negative angle –2𝜋 / 3, add 2𝜋 to obtain a coterminal angle. Student guided practice O Do problems 25 and 26 in your book page 261 Degree Measure O Definition: A degree is a unit of angle measure that is equivalent to the rotation in 1/360th of a circle. O Because there are 360° in a circle, and we now know that there are also 2π radians in a circle, then 2π = 360°. O 360° = 2π radians 2π radians= 360° O 180° = π radians π radians = 180° O 1° = 𝜋 radians 180 1 radian= 180/𝜋 Degree Measure O To convert radians to degrees, multiply by 𝜋 180 . O To convert degrees to radians, multiply by O 180/𝜋. Example O Example: Convert 120° to radians. O Example: Convert -315° to radians Example O Example: Convert 5𝜋 6 to degrees. O Convert 7 to degrees. Student guided practice O Do odd problems form 55-65 in your book page 262 Acute and Obtuse O An acute angle has a measure between O 0 𝜋 and . 2 O (or between 0° and 90°.) O An obtuse angle has a measure between 𝜋 2 and π (or between 90° and 180°.) Example O Example: Find the supplement and complement of 𝜋/5. Arc Length O Because we already know that with radian measure θ =r /s, O where s is the arc length, then s = r θ. Example O Example: Find the length of the arc that subtends a central angle with measure 120° in a circle with radius 5 inches. Example O A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240 degrees, as shown in Figure 4.15. Student Guided practice O Do problem 93 and 94 in your book page 263 Linear and angular speed O Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is O Linear speed = arc length/ time= s/t 𝑟𝜃 = 𝑡 Linear and angular speed O Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is Angular speed = central angle/ time=𝜃/ t. Example O Example: The circular blade on a saw rotates at 2400 revolutions per minute. O (a) Find the angular speed in radians per second. O (b) The blade has a diameter of 16 inches. Find the linear speed of a blade tip. Area of a sector Homework O Do problems 27,28,45,46,51,52,56,58,79,85 O In your book page 261 and 262 Closure O Today we learned about radian and degree measure O Next class we are going to learn about the unit circle