Warm Up Write answers in reduced pi form. Convert 8𝜋 9 to degrees Convert -2560̊ to radians Find the arc length The square has side lengths 14. The two curves are each 1 4 of a circle with radius 14. Find the area of the shaded region. 𝟗𝟖𝝅−𝟏𝟗𝟔

Section 7-3 The Sine and Cosine Functions Objective: To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

𝑠𝑖𝑛𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑦 𝑟 r  𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑥 𝑟 cos 𝜃=

Example 1 If the terminal ray of an angle θ in standard position passes through (-3, 2), find sin θ and cos θ. Solution: On a grid, locate (-3,2). Use this point to draw a right triangle, where one side is on the x-axis, and the hypotenuse is line segment between (-3,2) and (0,0). Start of Day 2

𝑟= −3 2 + 2 2 𝒙=−𝟑 𝒓= 𝟏𝟑 𝐲=𝟐 = 2 13 13 = 2 13 𝑠𝑖𝑛𝜃= 𝑦 𝑟 = −3 13 13 = 2 13 13 = 2 13 𝑠𝑖𝑛𝜃= 𝑦 𝑟 = −3 13 13 = −3 13 𝑐𝑜𝑠𝜃= 𝑥 𝑟 𝑟= −3 2 + 2 2 𝒙=−𝟑 𝒓= 𝟏𝟑 𝐲=𝟐

Example 2 If the 𝑠𝑖𝑛𝜃=− 5 13 , what quadrant is the angle in?

= 12 13 𝑐𝑜𝑠𝜃= 𝑥 𝑟 𝐲=−𝟓 𝒓=𝟏𝟑 𝑥= 13 2 − −5 2 𝒙=𝟏𝟐 𝒙=±𝟏𝟐 4th Quadrant, so

𝑠𝑖𝑛𝜃= cos 𝜃= 𝑦 𝑟 𝑥 𝑟 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑎𝑑𝑗 ℎ𝑦𝑝 = r  𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑥 𝑟 cos 𝜃= When the radius =1 on the unit circle, 𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦 𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥

Unit Circle The circle x2 + y2 = 1 has radius 1 and is therefore called the unit circle. This circle is the easiest one with which to work because sin θ and cos θ are simply the y- and x-coordinates of the point where the terminal ray of θ intersects the circle. When the radius =1 on the unit circle, 𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦 𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥

1 1 2 1 2

II I III IV (−,+) (+,+) (−,−) (+,−) On Your Unit Circle: Label the quadrants. Note the positive or negative x and y values in each quadrant. (cos, sin) (cos, sin) (−,+) (+,+) II I III IV (−,−) (+,−) (cos, sin) (cos, sin)

You can determine the exact value of sine and cosine for many angles on the unit circle. 1 -1 − 3 2 2 2 Find: sin 90° sin 450° cos (-π) sin (− 2𝜋 3 ) cos -315° Refer to graph file “UC Quadrantal Angles”

Example 3 Solve sin θ = 1 for θ in degrees and radians.

Multiple solutions to trig equations

Expressing multiple solutions to trig equations Degrees: 𝜃=90˚±360𝑛 𝜃= 𝜋 2 ±2𝑛𝜋 Radians: Where n can be any integer value

Repeating Sin and Cos Values For any integer n, 𝑠𝑖𝑛 (𝜃 ± 360°𝑛) = 𝑠𝑖𝑛⁡𝜃 𝑐𝑜𝑠 (𝜃 ± 360°𝑛) =𝑐𝑜𝑠⁡𝜃 𝑠𝑖𝑛 (𝜃 ±2𝜋𝑛) = 𝑠𝑖𝑛⁡𝜃 𝑐𝑜𝑠 (𝜃 ±2𝜋𝑛) =𝑐𝑜𝑠⁡𝜃 The sine and cosine functions are periodic. They have a fundamental period of 360˚ or 2 radians.

In-class practice part 1 P271 class exercises 1-9

Section 7-4 Evaluating & Graphing Sine and Cosine Objective: To use reference angles and the unit circle to find values of the sine and cosine functions.

The reference angle is always less than 90˚ The smallest angle that the terminal side of a given angle makes with the x-axis. The reference angle is always less than 90˚

Reference Angles 60 ̊ Locate 5𝜋 4 on your unit circle. 𝝅 𝟒 How far is 5𝜋 4 from the x-axis? 𝝅 𝟒 is the reference angle. Locate -240 ̊on your unit circle. How far is -240 ̊from the x-axis? 60 ̊ 60 ̊ is the reference angle.

Finding the reference angle, , for angle 𝜽. 0<𝜽< 360˚(2𝜋) 𝜽   𝜽 Quadrant I: =𝜽 𝜽 𝜽 Quadrant II: =𝟏𝟖𝟎−𝜽 =𝝅−𝜽   Quadrant III: =𝜽−𝟏𝟖𝟎 =𝜽−𝝅 Quadrant IV: =𝟑𝟔𝟎−𝜽 =𝟐𝝅−𝜽

Reference Angle,  The reference angle for 60˚ is 60˚

Reference Angle,  The reference angle for 240˚ is 60˚

The reference angle is measured from the terminal side of the original angle to the x-axis (not the y-axis).

30˚ 150˚ 210˚ 330˚ Name 4 angles between 0˚ and 360˚ that have a reference angle of 30˚. 150˚ 30˚ 210˚ 330˚

Draw each angle, then Find the reference angle Reference Angles 𝟎<𝜶< 𝝅 𝟐 Draw each angle, then Find the reference angle 1. 24° 2. −37° 3. 228.4° 4. −155° 5. −350° 6. −543° 7. 5𝜋 3 8. − 7𝜋 6 9. 11𝜋 6 10. 15𝜋 4 11. 𝜋 3 12. − 𝜋 4 24° 37° 48.4° 25° 10° 3° 𝝅 𝟑 𝝅 𝟔 𝝅 𝟔 𝝅 𝟒 𝝅 𝟑 𝝅 𝟒

Classwork In class practice part II Section 7.3 & 7.4 Practice Worksheet

Homework Suggested practice problems 7.3: p272: 1-19all, 33-39 odd this is a lot of problems, but they are almost all, answer with the unit circle. 7.4: p278: 1-17odd