Arc Length and Radians DO NOW 3/23: Mr. McLaren wants a 60 degree slice of a pizza with a radius of 15 inches. How many inches will the crust of his slice be? Arc Length and Radians Agenda 1. Do Now 2. Quiz Review 3. What is a radian? 4. Converting to radians 5. Debrief

Quiz Review DIAGRAMS AND EQUATIONS FIRST!

What is a radian? A radian is the measure around a circle that intercepts an arc the same length as the radius. If the radius=1, then the C = 2𝜋r is 2𝜋(1) or just 2𝜋. So in radians we say 360 degrees = 2𝜋, 180 degree = 1𝜋, 90 degrees = 𝜋 2 ….and so on.

Investigation: Radians Arc Length Radius Arc Length/Radius

How to convert radians to degrees To convert radians to degrees is easy! Just remember that 360° = 2𝜋! To convert degrees to radians (or any unit conversion): 𝜃° 1 𝑥 2𝜋 360° = 𝜃°2𝜋 360° Reduce and the resulting measure in terms of pi will be your radians. To convert the other way, you would just need to flip your fractions. 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 1 𝑥 360° 2𝜋 = 360°(𝑟𝑎𝑑) 2𝜋 **MUST BE IN NOTES!**

Independent Practice/Debrief How are degrees and radians related? What is the measure of 45 degrees in radians?

Radians and Unit Circle Do Now 3/24: What is the measure of 1 radian in degrees? Learning Target: I can create a unit circle and use it to convert degrees to radians. Agenda 1. Do Now 2. HW Review 3. Create a unit circle 4. Label degrees/radians 5. Debrief

HW Review: Radians  Degrees

Create a Unit Circle Use the record to trace a circle on the coordinate grid paper. Starting with 0 on the positive x axis, use a protractor to mark off every 15 degrees (all the way to 360!) Convert each angle measure from degrees to radians and label these as well.

Debrief What patterns do you notice in your unit circle? How can these patterns be related to trigonometry?

Find the length of the segment. Equation of a Circle Do Now 3/25: Find the length of the segment. Learning Target: I can compare/contrast Pythagorean theorem and distance formula to derive the equation of a circle on the coordinate grid. Agenda 1. Do Now 2. Calculating the Radius of your Circle 4. Equation of Circle 5. Debrief

Unit Circle Choose a coordinate that lies on your circle. Connect that point with the center point to create a radius. Draw a right triangle using the radius as the hypotenuse. Find the lengths of the legs of your right triangle. Use the lengths of your legs to find the lengths of your radius. a2 + b2 = c2

Equation of a Circle (6, 6) (2, 3) (x1 - x2)2 + (y1 - y2)2 = d2 What if we have limited information? For example, only are given the coordinates of the center and the point on the edge? Center (0, 0) and point (4, 8) Center (3, 4) and point (8, 11) (6, 6) (2, 3) (x1 - x2)2 + (y1 - y2)2 = d2

Equation of a Circle a2 + b2 = c2 (x1 - x2)2 + (y1 - y2)2 = d2 How can we use these two equations to help us derive a formula for the radius of a circle with center point (h, k) and a point (x, y) on the circle? (x-h)2 + (y-k)2 = r2 Equation of a Circle **MUST BE IN NOTES!**

Equation of a Circle: Debrief/Exit Ticket Write the equation of a circle with center point (2, 0) and a radius of 10. Write the equation of a circle with center point (3, 6) and contains the point (0, -2)

Circle Equation and Trigonometry DO NOW 3/26: Identify the center and radius for the circle equation below. Then sketch the graph. Learning Target: I can use the coordinate grid to relate circles and right triangle trigonometry Agenda 1. Do Now 2. Circle Equation Practice 3. Unit Circle and Trig 4. Debrief

Circle Equation Practice

QUICK Trig Review 3 trig functions (SOH CAH TOA) Sine(𝜃) = Opp/Hyp Cosine(𝜃) = Adj/Hyp Tangent(𝜃) = Opp/Adj Using Calculators with Trig: To find missing side, set up proportion and cross multiply, use (sin/cos/tan) button. To find missing angle (𝜃), use 2nd function (sin/cos/tan) and multiply by the ratios of the sides.

Circles and Trigonometry x y On your circle, choose a coordinate and draw a right triangle. Label the opposite, adjacent, and hypotenuse of that right triangle, using the central angle as theta. What is another way we can describe the opposite side of this triangle on the coordinate grid? The adjacent? What is the hypotenuse in the circle? How can we represent the trigonometric ratios (sine, cosine, tangent) using a circle?

Exit Ticket Choose a coordinate on your circle; draw a right triangle to that coordinate; use trigonometry to find the measure of the angle; then convert the angle to radians. BONUS: Write the equation of your circle and use your coordinate to prove the length of the radius.

Debrief How can we use trigonometry to discuss angles in circles? Learning Target: I can use the coordinate grid to relate circles and right triangle trigonometry How can we use trigonometry to discuss angles in circles? What do the ratios of sine, cosine, and tangent represent in terms of a circle?