M3U7D2 Warm Up Collect INTERIMS!!! (x+4)2 (x-7)2 c = 36 (x+6)2 1. Factor 2. Factor What would the value of c that makes a perfect square. Then write as a perfect square. (x+4)2 (x-7)2 c = 36 (x+6)2 Collect INTERIMS!!!

HW Check Document Camera

U7D2 Unit Circle OBJ: Build an understanding of trig functions by using tables, graphs and technology to represent the cosine and sine functions. Interpret the sine function as the relationship between the radian measure and an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate. Interpret the cosine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.

WRITE THIS DOWN on pp. 7 The Unit Circle Definition: A circle centered at the origin with a radius of exactly one unit. Let’s develop! (0, 1) |-----1-------| (-1,0) (0 , 0) (1,0) (0, -1) 4

WRITE THIS DOWN WITH THE PREVIOUS SLIDE What are the angle measurements of each of the four angles we just found? π/2 90° 0° 2π 180° 360° π 270° 3π/2 5

The Unit Circle WATCH Let’s look at an example In order to determine the sine and cosine we need a right triangle. The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle. 1 30 -1 1 -1

WATCH The Unit Circle Create a right triangle, using the following rules: The radius of the circle is the hypotenuse. One leg of the triangle MUST be on the x axis. The second leg is parallel to the y axis. 1 30 -1 1 -1 60 2 1 Remember the ratios of a 30-60-90 triangle- 30  

The Unit Circle WATCH 2 60 1 1 30 P X- coordinate 30 -1 1   P X- coordinate 30   -1 1 Y- coordinate   -1  

Angles and the Unit Circle WATCH Find the cosine and sine of 135°. From the figure, the x-coordinate of point A  is – , so cos 135° = – ,or about –0.71. 2 Use a 45°-45°-90° triangle to find sin 135°. opposite leg = adjacent leg 0.71   Simplify. =    Substitute. 2 The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 135° –0.71 and sin 135° 0.71.

Angles and the Unit Circle WATCH Find the exact values of cos (–150°) and sin (–150°). Step 1:  Sketch an angle of –150° in standard position. Sketch a unit circle. x-coordinate = cos (–150°) y-coordinate = sin (–150°) Step 2:  Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.)

Angles and the Unit Circle WATCH (continued) The triangle contains angles of 30°, 60°, and 90°. Step 3: Find the length of each side of the triangle. hypotenuse = 1 The hypotenuse is a radius of the unit circle. shorter leg = The shorter leg is half the hypotenuse. 1 2 1 2 3 longer leg = ∙ 3 = The longer leg is 3 times the shorter leg. 3 2 1 Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so cos (–150°) = – , and sin (–150°) = – .

THEREFORE: The Unit Circle WRITE THIS DOWN THEREFORE: The Unit Circle A circle with radius of 1 has the Equation x2 + y2 = 1

Switch to document Camera Develop Unit Circle Measures Demonstrate how to find angle measure in radians in Quadrant I Expand to all four quadrants Demonstrate how to find since and cosine

The Unit Circle with Radian Measures COMPLETE ON HANDOUT The Unit Circle with Radian Measures

Some common radian measurements These are the Degree expressed in Radians

Summarized The Trig functions

SOH CAH TOA Sine is Opp/Hyp Cosine is Adj/Hyp Tangent is Opp/Adj FYI…M2 REMINDERS ??? SOH CAH TOA Sine is Opp/Hyp Cosine is Adj/Hyp Tangent is Opp/Adj Cosecant is 1/Sine Secant is 1/Cosine Cotangent is 1/Tangent

SOH CAH TOA

Lets find the Trig functions of Think where this angle is on the unit circle.

Test Corrections due Thursday 12/14/17 Classwork Test Corrections due Thursday 12/14/17 pp. 8 Homework pp. 9&10 AND complete the unit circle games (QUIZ) on the next slide– email me a screenshot of your score by FRIDAY 12/15/17!

Study Aids Terminal Side Coterminal Degrees to radians flashcards Degrees, Radians and Coordinate points flashcards Angles of the Unit Circle – Radians QUIZ Angles of the Unit Circle - Degrees QUIZ