Math III Accelerated Chapter 13 Trigonometric Ratios and Functions 1

Warm Up 13.2  Write the expression in simplest form. 2

13.2 Define General Angles and Use Radian Measure  Objective:  Use general angles that may be measured in radians. 3

Angles in Standard Position  In the coordinate plane, an angle can be formed by fixing one ray, called the _______________ side, and rotating the other ray, called the ____________________ side, about the ____________________.  An angle is in standard position if its vertex is at ___________ and its initial side lies on the positive ____________________. 4

Positive and Negative Angles  A positive angle opens _________________.  A negative angle opens _________________. 5

Example 1 a. Draw a 405 ˚ angle in standard position. b. Draw a –65 ˚ angle in standard position. 6

Coterminal Angles  Angles that share the same terminal side are called coterminal angles.  Coterminal angles can be found by adding (or subtracting) multiples of 360˚ to (from) the given angle. 7

Example 2  Find one positive and one negative angle that are coterminal with 210˚. Sketch your results. 8

Checkpoints 1 & 2  Draw an angle with the given measure in standard position. Find one positive and one negative coterminal angle. a. 485 ° b. –75° 9

Radian Measure  Angles can be measured either in degrees or in radians. 10

Converting Between Degrees and Radians  Degrees to Radians Radians = Degrees  Radians to Degrees Degrees = Radians 11

Example 3  Convert : a. 315˚ to radians b. π /6 radians to degrees 12

Checkpoints 3 & 4  Convert the degree measure to radians or the radian measure to degrees. a. 200˚ b. π /5 13

Sectors of Circles  A sector is a region of a circle that is bounded by two radii and an arc of the circle.  The central angle θ of a sector is the angle formed by the two radii. 14 θ r r

Arc Length and Area of a Sector  The arc length s and area A of a sector with radius r and central angle θ (measured in radians) are:  Arc Length s = r θ  Area A = ½ r 2 θ 15 θ r s

Example 4  Find the arc length and area of a sector with a radius of 15 inches and a central angle of 60 ˚. 16

Checkpoint 5  Find the arc length and area of a sector with a radius of 5 feet and a central angle of 75 ˚. 17

Homework 13.2  Practice