1. 9.4 Trigonometry: Cosine Ratio
Geometry
9.4 Trigonometry: Cosine Ratio

2. 9.4 Cosine Ratio Cosine Ratio, Secant Ratio, and Inverse Cosine
Objectives
Use the cosine ratio in a right triangle to solve for unknown side lengths.
Use the secant ratio in a right triangle to solve for unknown side lengths.
Relate the cosine ratio to the secant ratio.
Use the inverse cosine in a right triangle to solve for unknown angle measures.
SOH – CAH – TOA

3. 9.4 Problem 1 Making a Tower Stable
Together 1-2
The COSINE (COS) of an acute angle in a right triangle is the ratio of the length of the side that is adjacent to the angle to the length of the hypotenuse.
Collaborate 3-6 (4 Minutes)
cos 53= cos 37=0.8
cos 48= cos 42=0.74
cos 60= cos 30=0.87

4. 9.4 Problem 1 Making a Tower Stable
Together 7.

5. 9.4 Problem 1 Making a Tower Stable
Collaborate 8-12 (8 Minutes)
Trigonometric Ratios
sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔
SOH-CAH-TOA

6. 9.4 Problem 1 Making a Tower Stable8.

7. 9.4 Problem 1 Making a Tower Stable

8. 9.4 Problem 1 Making a Tower Stable
10.

9. 9.4 Problem 1 Making a Tower Stable
11.
12.

10. 9.4 Problem 1: Making a Tower Stable
Together #13
tan 𝜃 = sin 𝜃 cos 𝜃
tan 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝 𝐴𝑑𝑗 𝐻𝑦𝑝
tan 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝 𝑥 𝐻𝑦𝑝 𝐴𝑑𝑗
tan 𝜃 = 𝑂𝑝𝑝 𝐴𝑑𝑗

11. 9.4 Problem 2 Secant Ratio
The secant of an angle is the reciprocal of the cosine of an angle.
The SECANT (SEC) 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 adjacent to the angle.
We can always use the cosine function.

12. 9.4 Problem 2 Secant Ratio On the calculator There is no secant button
SEC is the reciprocal of COS
sec 𝜃 = 1 cos 𝜃
Example
𝐹𝑖𝑛𝑑 sec 35 𝑜
= 1 cos 35 𝑜 ≈1.22

13. 9.4 Problem 3 Inverse Cosine
The inverse cosine (arc cosine or 𝑐𝑜𝑠 −1 ) is the measure of an acute angle whose cosine is x.
The relationship of sides used is adjacent to hypotenuse.
The calculator has an inverse cosine function.
Only used to find the missing angle.

14. 9.4 Problem 3 Inverse Cosine
Collaborate 1-4 (4 Minutes)

15. 9.4 Problem 3 Inverse Cosine

16. 9.4 Problem 3 Inverse Cosine

17. 9.4 Problem 3 Inverse Cosine

18. Example of SOH-CAH-TOA
Find the measure of angle A using all 3 trig function
Collaborate: 1 Minutes
sin 𝐴 = 3 5
𝐴=𝑠𝑖𝑛 − ≈ 36.9 𝑜 5
cos 𝐴 = 4 5
𝐴=𝑐𝑜𝑠 − ≈ 36.9 𝑜 3
tan 𝐴 = 3 4
𝐴=𝑡𝑎𝑛 − ≈ 36.9 𝑜 A 4

19. Hypotenuse X
Adjacent
cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos 65 = 𝑥 12
𝑥=12∙ cos 65
𝑥≈5.07 𝑓𝑒𝑒𝑡

20. Use the given diagram to find the missing sideX
28 𝑜
100 ft
cos 28 = 100 𝑥
𝑥 cos 28 =100
𝑥= 100 cos 28 ≈113.3 𝑓𝑒𝑒𝑡
Shortcut
cos 28 = 100 𝑥
𝑆𝑤𝑖𝑡𝑐ℎ
𝑥= 100 𝑐𝑜𝑠 28 ≈113.3 𝑓𝑒𝑒𝑡

21. Use the given diagram to find the missing side
100 ft
28 𝑜 X
cos 28 = 𝑥 100
100∗ cos 28 =𝑥
𝑥≈88.29 𝑓𝑒𝑒𝑡

22. Formative Assessment SOH-CAH-TOA Skills Practice 9.4
Problem Set Pg (1-43) odd
cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sec 𝜃 = 1 cos 𝜃
SOH-CAH-TOA
Check the MODE on the calculators
The MODE must be in DEGREES