Trigonometry 5.1 Radian & Degree Measure

Trigonometry Vocabulary 1.) A Greek letter that is used when labeling angles in trigonometry ( α ) alpha 2A.) An angle that has an initial side that corresponds to the positive x axis is said to be in this position. standard 2D.) Synonymous with intercepts. subtends 3.) A Greek letter used to name angles (β). beta 4.) The typical type of angle measure used in Trig, Precalculus, and Calculus. Radians 5.) A greek letter θ used to represent angles. theta 6.) There are 3600 of these in a degree. seconds

7.) Negative angles are rotated clockwise. 8.) The central angle of a circle is formed by the center of the circle and two rays that intersect the circle. 9.) The arc length = the radius when θ = 1 radian. 10.) Real world applications of trig include: sound waves, light rays, planetary orbits, pendulum, vibrating strings, and orbits of atomic particles. 11.) 2πr circumference 12.) Positive angles are rotated in this direction. counterclockwise 13.) The measure of an angle is determined by the amount of rotation from initial to terminal side.

14.) If θ is the angle (in radians) corresponding to the arc length s, then the angular speed of the particle is: angular speed = θ/t. 15.) Determined by rotating a ray about its endpoint. angle 16.) A minute is 1/60th of a degree. 17.) The “measurement of triangles.” Trigonometry 18.) There are 360 of these in a circle. degrees 19.) The point of an angle. vertex 20.) Two angles that have the same terminal ray. coterminal 21.) The ray that starts the angle is referred to as this. initial side

Crossword Solution

Angles of Rotation Standard Position Positive Angle Negative Angle Vertex Θ = Θ = Initial Side Terminal Side Rotations

Radian Measures A radian is the measure of an angle at which the radius is equal to the arc length. Formulas Relating Angle (  ), Arc Length(s), and Radius(r) s = r   = r = Note:  is in radians, not degrees

Arc length around the circle = circumference s = 2πr 1 revolution = 2π ≈ 6.28 radians 1 radian = 360 degrees ÷ 2π radians ≈ 57.3 o

Portions of a Revolution (examples) 1 revolution = = 2π radians ½ revolution = = π radians ¼ revolution = 90 0 = radians 1/6 revolution = 60 0 = radians

Activity Complete common angle chart Board practice on angle rotations

Circle Divided Into 8 Parts

Circle Divided Into 12 Parts

Coterminal Angles Angles with the same terminal ray. To find them, add multiples of or 2π.

Coterminal Examples Find a positive and negative coterminal angle for the following. a b = – =

Complimentary & Supplementary Angles Complimentary angles – sum of 90 0 or C: 90 0 – 39 0 = S: – 39 0 = Supplementary angles – sum of or π C: S:

Degree and Radian Conversions

Examples 1. Convert to radians 2. Convert to degrees.

Activity Board practice on... Coterminal angles Supplementary/Complimentary angles Radian & Degree conversions End Day 1 of Lesson 5.1

Degree, Minutes, and Seconds For more accuracy, portions of degrees are divided up into minutes and seconds There are 60 minutes in a degree and 60 seconds in a minute There are 3600 seconds in a degree ’ 48”

Applications of s = rθ A circle has a radius of 3 inches. Find the arc length subtended by a central angle of (Remember θ must be in radians) s = rθ s 3 in.

You Try! If a circle has an arc length of 9 cm, find the radius when the central angle is cm r

Activity Complete the 5.1 Practice Worksheet Discuss answers when finished Hwk. Pg. 333 #37 – 57 odd Pg. 333 #3 – 35 (x4) optional