1. 9.3 Trigonometry: Sine Ratio
Geometry
9.3 Trigonometry: Sine Ratio

2. 9.3 The Sine Ratio Sine Ratio, Cosecant Ratio, and Inverse Sine
Objectives
Use the sine ratio in a right triangle to solve for unknown side lengths.
Use the cosecant ratio in a right triangle to solve for unknown side lengths.
Relate the sine ratio to the cosecant ratio.
Use the inverse sine in a right triangle to solve for unknown angle measures.
SOH – CAH – TOA

3. 9.3 Problem 1 Fore!
Collaborate 1-3 (3 Minutes)

4. 9.3 Problem 1 Fore!
What affect does the angle of the club face have on the ball?
Ratio of Opposite to Hypotenuse
21 𝑜 : ≈0.36
35 𝑜 : ≈0.57
39 𝑜 : ≈0.63
What happens to the ratio as the angle gets larger?

5. 9.3 Problem 1 Fore!
The SINE (SIN) of an acute angle in a right triangle is the ratio of the length of the side that is opposite the angle to the length of the hypotenuse.
Collaborate 4-10 (10 Minutes)
Sine value refers to the sine of the angle measure.
There are no missing pieces to solve for. We are “proving” that the ratio is equal to the sine of the angle.
CALCULATOR MODE MUST BE IN DEGREES

6. 9.3 Problem 1 Fore! #5 sin 21 = 19.3 54 ≈0.36 sin 69 = 50.4 54 ≈0.93

7. 9.3 Problem 1 Fore! 9.

8. 9.3 Problem 2 Cosecant Ratio
The cosecant of an angle is the reciprocal of the sine of an angle.
The COSECANT (CSC) of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side that is opposite the angle.
We can always use the sine function.

9. 9.3 Problem 2 Cosecant Ratio
On the calculator
There is no cosecant button
CSC is the reciprocal of SIN
csc 𝜃 = 1 sin 𝜃
Example
𝐹𝑖𝑛𝑑 c𝑠𝑐 35 𝑜
= 1 sin 35 𝑜 ≈1.74

10. 9.3 Problem 3 Inverse Sine
The inverse sine (arc sine or 𝑠𝑖𝑛 −1 ) is the measure of an acute angle whose sine is x.
The relationship of sides used is opposite to hypotenuse.
The calculator has an inverse sine function.
Only used to find the missing angle.

11. 9.3 Problem 3 Inverse Sine
Collaborate 1-4 (5 Minutes)

12. 9.3 Problem 3 Inverse Sine

13. 9.3 Problem 3 Inverse Sine

14. 9.3 Problem 3 Inverse Sine

15. 9.3 Problem 3: Inverse Sine Together #5 sin 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝
𝜃= 𝑠𝑖𝑛 −
𝜃≈ 3.99 𝑜
We have the opposite side
Which side do we still need?
The hypotenuse is, on average, 56 meters

16. 𝜽 Hypotenuse Opposite sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sin 𝜃 = 4 14
𝜃= 𝑠𝑖𝑛 −
𝜃≈ 16.6 𝑜

17. Use the given diagram to find the missing sideX
100 ft
28 𝑜
sin 28 = 100 𝑥
𝑥 sin 28 =100
𝑥= 100 sin 28 ≈213 𝑓𝑒𝑒𝑡
Shortcut
sin 28 = 100 𝑥
𝑆𝑤𝑖𝑡𝑐ℎ
𝑥= 100 𝑠𝑖𝑛 28 ≈213 𝑓𝑒𝑒𝑡

18. Use the given diagram to find the missing side
100 ft X
28 𝑜
sin 28 = 𝑥 100
100∗ sin 28 =𝑥
𝑥≈46.95 𝑓𝑒𝑒𝑡

19. Formative Assessment SOH-CAH-TOA Skills Practice 9.3
Problem Set Pg (1-43) Odd
sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
csc 𝜃 = 1 sin 𝜃
SOH-CAH-TOA
Check the MODE on the calculators
The MODE must be in DEGREES