1. Geometry 9.5 Trigonometric Ratios
9.5 Warmup
1. Can these 3 side lengths 2. Find tan B. make a Triangle? If yes, classify the triangle. 7, 9, Find x & y. X
February 22, 2019
Geometry 9.5 Trigonometric Ratios

2. 9.4 & 9.5 Trigonometric Ratios
Geometry
9.4 & 9.5 Trigonometric Ratios

3. Geometry 9.5 Trigonometric Ratios
Essential Question
How is a right triangle used to find the sine, cosine, and tangent of an acute angle?
February 22, 2019
Geometry 9.5 Trigonometric Ratios

4. Geometry 9.5 Trigonometric Ratios
Goals
Find the sine, cosine, and tangent of an acute angle.
Solve problems using trigonometric ratios.
February 22, 2019
Geometry 9.5 Trigonometric Ratios

5. Geometry 9.5 Trigonometric Ratios
Right Triangle
Hypotenuse
Opposite A
Adjacent
February 22, 2019
Geometry 9.5 Trigonometric Ratios

6. Geometry 9.5 Trigonometric Ratios
Sine and Cosine Ratios
February 22, 2019
Geometry 9.5 Trigonometric Ratios

7. Geometry 9.5 Trigonometric Ratios
Trig Ratios A
February 22, 2019
Geometry 9.5 Trigonometric Ratios

8. Geometry 9.5 Trigonometric Ratios
Memory Aid
Sine is Opposite over Hypotenuse.
Cosine is Adjacent over Hypotenuse.
Tangent is Opposite over Adjacent.
SOH CAH TOA
February 22, 2019
Geometry 9.5 Trigonometric Ratios

9. Writing Ratios SOH CAH TOA3 4 5 A B
February 22, 2019
Geometry 9.5 Trigonometric Ratios

10. Geometry 9.5 Trigonometric Ratios
Example 1
Find sin J, cos J, and tan J. Write each answer as a fraction and as a decimal rounded to four places.
24 40 = 3 5 = .6
sin J =
cos J =
tan J =
32 40 = 4 5 = .8
24 32 = 3 4 =.75
Geometry 9.5 Trigonometric Ratios
February 22, 2019

11. Geometry 9.5 Trigonometric Ratios
Your Turn
Find sin K, cos K, and tan K. Write each answer as a fraction and as a decimal rounded to four places.
32 40 = 4 5 = .8
sin K =
cos K =
tan K =
24 40 = 3 5 = .6
32 24 = 4 3 =1.3333
February 22, 2019
Geometry 9.5 Trigonometric Ratios

12. Sine and Cosine of Complementary Angles
February 22, 2019
Geometry 9.5 Trigonometric Ratios

13. Geometry 9.5 Trigonometric Ratios
Example 2
Find the cosine that is equivalent to the given sine.
sin 83°
sin 34°
sin 57°
February 22, 2019
Geometry 9.5 Trigonometric Ratios

14. Geometry 9.5 Trigonometric Ratios
Your Turn
Find the sine that is equivalent to the given cosine.
cos 26°
cos 73°
cos 45°
February 22, 2019
Geometry 9.5 Trigonometric Ratios

15. Geometry 9.5 Trigonometric Ratios
Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle. 2x
Use a special right triangle to find the sine and cosine of a 60° angle. x
sin 60= 𝑜𝑝𝑝 ℎ𝑦𝑝 sin 60= 3 2 Check: sin 60≈.8660
x 3
cos 60= 𝑎𝑑𝑗 ℎ𝑦𝑝
cos 60= 1 2
Check: sin 60≈.5
February 22, 2019
Geometry 9.5 Trigonometric Ratios

16. Geometry 9.5 Trigonometric Ratios
Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle. 2x
Use a special right triangle to find the sine and cosine of a 30° angle. x
sin 30= 𝑜𝑝𝑝 ℎ𝑦𝑝
sin 30= 1 2
Check: sin 30≈.5
x 3
cos 30= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 30= 3 2 Check: cos 30≈.8660
February 22, 2019
Geometry 9.5 Trigonometric Ratios

17. Geometry 9.5 Trigonometric Ratios
Using Special Right Triangles to find sine and cosine of a 30, 60 or 45 angle.
x 2
Use a special right triangle to find the sine and cosine of a 45° angle. x
sin 45= 𝑜𝑝𝑝 ℎ𝑦𝑝
sin 45=
sin 45=
Check: sin 45≈.7071
cos 45= 𝑎𝑑𝑗 ℎ𝑦𝑝
cos 45=
cos 45=
Check: cos 45≈.7071 x ∙ ∙
February 22, 2019
Geometry 9.5 Trigonometric Ratios

18. Geometry 9.5 Trigonometric Ratios
Calculators
Remember to make sure your calculator is in DEGREE mode.
Always use four decimal places of accuracy when using trig functions.
February 22, 2019
Geometry 9.5 Trigonometric Ratios

19. Geometry 9.5 Trigonometric Ratios
Find these values:
sin 15
cos 45
tan 45
cos 80
sin 10
tan 5
cos 60
.2588
.7071 1
.1736
.0875 .5
February 22, 2019
Geometry 9.5 Trigonometric Ratios

20. Geometry 9.5 Trigonometric Ratios
Solving Triangles
To solve a Right Triangle, all we need is ONE acute angle and ONE side.
Carefully analyze the given information.
Decide what you are trying to find.
Ask: How do you use the given information to solve the problem?
WRITE AN EQUATION. (SOH CAH TOA)
Solve.
February 22, 2019
Geometry 9.5 Trigonometric Ratios

21. Geometry 9.5 Trigonometric Ratios
Example 3 Find x.
From the 28 angle, x is the ?
Opposite side,
and 15 is the
Hypotenuse.
What trig ratio is this?
Sine (SOHCAHTOA)
28 x 15
February 22, 2019
Geometry 9.5 Trigonometric Ratios

22. Geometry 9.5 Trigonometric Ratios
Example 3 Find x.
Write the equation and solve.
28 x 15
February 22, 2019
Geometry 9.5 Trigonometric Ratios

23. Geometry 9.5 Trigonometric Ratios
Example 4 Find y.
Write the equation and solve. 56
31 y
February 22, 2019
Geometry 9.5 Trigonometric Ratios

24. Geometry 9.5 Trigonometric Ratios
Example 5 Find x & y.
tan 26= 𝑜𝑝𝑝 𝑎𝑑𝑗 tan 26= 𝑥 tan 26=𝑥 𝑦≈6.8283
cos 26= 𝑎𝑑𝑗 ℎ𝑦𝑝 cos 26= 14 𝑦 y cos 26=14 𝑦= 14 𝑐𝑜𝑠26 𝑦≈15.6
26 y 14 x
February 22, 2019
Geometry 9.5 Trigonometric Ratios

25. Geometry 9.5 Trigonometric Ratios
Your Turn
You are skiing on a mountain. You start at an altitude of 8400 feet and ski down to an altitude of The angle of depression is 21°. Find the distance x you ski down the mountain to the nearest foot. y
You ski about 3349 ft down the mountain.
February 22, 2019
Geometry 9.5 Trigonometric Ratios

26. Geometry 9.5 Trigonometric Ratios
Homework
February 22, 2019
Geometry 9.5 Trigonometric Ratios