1. 𝑥=4, 𝑦=5 𝑥=21, 𝑟=29 𝑟= 10 , 𝑥=1 𝑦=4 2 , 𝑥=4 Warm Up r y 𝒓= 𝟒𝟏 x 𝒚=𝟐𝟎
A right triangle has side lengths x, y, and r. Find the unknown length. r y
𝑥=4, 𝑦=5
𝑥=21, 𝑟=29
𝑟= 10 , 𝑥=1
𝑦=4 2 , 𝑥=4
The square has side lengths 14. The two curves are each of a circle with radius 14. Find the area of the shaded region.
𝒓= 𝟒𝟏 x
𝒚=𝟐𝟎
𝒚=𝟑
𝒓=𝟒 𝟑
𝟗𝟖𝝅−𝟏𝟗𝟔

2. Quiz 7.1 & 7.2 Degree & Radian Conversions Coterminal Angles
Arc Length of a Sector
Area of a Sector
Apparent Size

3. 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.

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

5. 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

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

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

8. = 12 13
𝑐𝑜𝑠𝜃= 𝑥 𝑟
𝐲=−𝟓
𝒓=𝟏𝟑
𝑥= − −5 2
𝒙=𝟏𝟐
𝒙=±𝟏𝟐
4th Quadrant, so

9. 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

10. 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

11. 1
1 2
1 2

12. 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)

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

14. Example 3 Degrees: 𝜃=90˚±360𝑛 Radians: 𝜃= 𝜋 2 ±2𝑛𝜋
Solve sin θ = 1 for θ in degrees and radians.
Degrees: 𝜃=90˚±360𝑛
Radians:
𝜃= 𝜋 2 ±2𝑛𝜋

15. 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.

16. Homework
Page 272
#1-27 odd, #33-41 odd