Entry Task. 13.3 Radian measure I can… change Degrees to Radians and radians to degrees change Degrees to Radians and radians to degrees Find the measure.

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Presentation transcript:

Entry Task

13.3 Radian measure I can… change Degrees to Radians and radians to degrees change Degrees to Radians and radians to degrees Find the measure of an intercepted arc

Definition of an angle An angle is made from two rays with a common initial point. In standard position the initial side is on the x axis

Positive angle vs. Negative angle Positive angles are Counter clockwise C.C.W. Negative angles are ClockwiseC.W.

Angles with the same initial side and terminal side are coterminal.

The measure of an angle is from initial side to terminal side Vertex at the origin (Center)

Definition of a Radian Radian is the measure of the arc of a unit circle. Unit circle is a circle with a radius of 1.

The quadrants in terms of Radians What is the circumference of a circle with radius 1?

The quadrants in terms of Radians What is the circumference of a circle with radius 1?

The quadrants in terms of Radians The circumference can be cut into parts.

The quadrants in terms of Radians The circumference can be cut into parts.

Find the Coterminal Angle Since equals 0. it can be added or subtracted from any angle to find a coterminal angle. Given

The “Magic” Proportion This proportion can be used to convert to and from Degrees to Radians. Degrees° 180° = r radians radians Example: Find the radian measure of an angle of 45°. Write a proportion. 45° 180° = r radians radians An angle of 45° measures about radians. Write the cross-products. 45 = 180 r Divide each side by 45.r = = 0.785Simplify. 4

The “Magic” Proportion This proportion can be used to convert to and from Degrees to Radians. Degrees° 180° = r radians radians Example: Find the radian measure of angle of -270°. Write a proportion. -270° 180° = r radians radians An angle of -270° measures about radians. Write the cross-products = 180 r Divide each side by 45.r = Simplify. 2 -3

Change 140º to Radians Change to degrees Usedegree to rads. Userads to degrees

Example = 390°Simplify. Find the degree measure of Write a proportion radians = d° = dWrite the cross-product d = Divide each side by An angle of radians measures 390°. 6 13

Example Find the degree measure of an angle of – radians. 2 3 = –270° An angle of – radians measures –270°. 2 3 – radians = – radians ° radians ° radians 1 90 Multiply by 180° radians.

Radian Measure Find the radian measure of an angle of 54°. 5 4° radians = 54° radians Multiply by radians. 180° radians=Simplify. An angle of 54° measures radians. 10 3

Draw the angle. Radian Measure Find the exact values of cos and sin. radians 3 3 radians = 60° Convert to degrees ° radians Complete a 30°-60°-90° triangle. The hypotenuse has length 1. radians 3 Thus, cos = 1212 and sin radians 3 =. 3 2 The shorter leg is the length of the hypotenuse, and the longer leg is 3 times the length of the shorter leg. 1212

Entry Task Sketch a Unit Circle Without looking fill in the degrees and radians for quadrant 1

How to use radians to find Arc length The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle. Circumference 360º In radians we use which equals Where S is the arc length

Radian Measure Use this circle to find length s to the nearest tenth. s = r  Use the formula. The arc has length 22.0 in. = 7Simplify. 22.0Use a calculator. = 6 Substitute 6 for r and for 

Find the length of the minor arc r = 9, θ = 215ºChanging to radians Arc length S

Homework P. 848 #26-34, 36, odds, 50