1. 7-5 and 7-6: Apply Trigonometric Ratios
Geometry Chapter 7
7-5 and 7-6: Apply Trigonometric Ratios

2. Warm-Up
Find the values of the variables in the following right triangles.
1.) )

3. Apply Trigonometric Ratios
Objective: Students will be able to solve for missing side lengths in right triangles using trigonometric ratios.
Agenda
Trigonometry
Sine
Cosine
Tangent

4. Trigonometry
A Trigonometric Ratio is the ratio of lengths of two sides of a right triangle.
You will use Trigonometric Ratios to find the measure of a side or of an acute angle in a right triangle.

5. Trigonometry 𝑩 𝑨 𝑪 Example: For ∆𝑨𝑩𝑪
The leg across from the marked angle is known as the Opposite Leg and the leg attached to the angle is known as the Adjacent Leg. Across from the right angle will always be the Hypotenuse. 𝑨 𝑩 𝑪
Hypotenuse
Opposite Leg
Adjacent Leg

6. Trigonometry 𝑩 𝑨 𝑪 Example: For ∆𝑨𝑩𝑪
The leg across from the marked angle is known as the Opposite Leg and the leg attached to the angle is known as the Adjacent Leg. Across from the right angle will always be the Hypotenuse. 𝑨 𝑩 𝑪
For Example:
𝑩𝑪 is the Leg Opposite to <𝑨
And
𝑨𝑪 is the Leg Adjacent to <𝑨
Hypotenuse
Opposite Leg
Adjacent Leg

7. The Sine Ratio Sine of <𝐴= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐴 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐵𝐶 𝐴𝐵
The first ratio is known as the Sine Ratio.
Sine of <𝐴= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐴 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐵𝐶 𝐴𝐵
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪

8. The Sine Ratio Sine of <𝐵= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐵 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐴𝐶 𝐴𝐵
The first ratio is known as the Sine Ratio.
Sine of <𝐵= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐵 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐴𝐶 𝐴𝐵
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪

9. Sine Examples From the given triangle, find 𝒔𝒊𝒏⁡𝑿 and 𝒔𝒊𝒏⁡𝒀. 𝟏𝟓 𝟖 𝟏𝟕 𝑿𝒁

10. Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Sine Examples
From the given triangle, find 𝒔𝒊𝒏⁡𝑿 and 𝒔𝒊𝒏⁡𝒀.
Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁

11. Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Sine Examples
From the given triangle, find 𝒔𝒊𝒏⁡𝑿 and 𝒔𝒊𝒏⁡𝒀.
Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐬𝐢𝐧 𝒙 = 𝟖 𝟏𝟕 ≈𝟎.𝟒𝟕𝟎𝟔 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁

12. Sine Examples From the given triangle, find 𝒔𝒊𝒏⁡𝑿 and 𝒔𝒊𝒏⁡𝒀.
Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐬𝐢𝐧 𝒙 = 𝟖 𝟏𝟕 ≈𝟎.𝟒𝟕𝟎𝟔 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁
Sine of 𝑌= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑌 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

13. Sine Examples From the given triangle, find 𝒔𝒊𝒏⁡𝑿 and 𝒔𝒊𝒏⁡𝒀.
Sine of 𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐬𝐢𝐧 𝒙 = 𝟖 𝟏𝟕 ≈𝟎.𝟒𝟕𝟎𝟔 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁
Sine of 𝑌= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑌 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐬𝐢𝐧 𝒚= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒

14. The Cosine Ratio
The second ratio is known as the Cosine Ratio.
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪
Cosine of <𝐴= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝐴 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐴𝐶 𝐴𝐵

15. The Cosine Ratio
The second ratio is known as the Cosine Ratio.
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪
Cosine of <𝐵= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝐵 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝐵𝐶 𝐴𝐵

16. Cosine Examples From the given triangle, find cos⁡𝑿 and cos⁡𝒀. 𝒀 𝟏𝟕 𝟖𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁

17. Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine Examples
From the given triangle, find cos⁡𝑿 and cos⁡𝒀.
Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁

18. Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine Examples
From the given triangle, find cos⁡𝑿 and cos⁡𝒀.
Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐜𝐨𝐬 𝒙= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁

19. Cosine Examples 𝐜𝐨𝐬 𝒙= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒
From the given triangle, find cos⁡𝑿 and cos⁡𝒀.
Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐜𝐨𝐬 𝒙= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁
Cosine of 𝑌= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑌 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

20. Cosine Examples 𝐜𝐨𝐬 𝒙= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒 𝐜𝐨𝐬 𝒚= 𝟖 𝟏𝟕 ≈𝟎.𝟒𝟕𝟎𝟔
From the given triangle, find cos⁡𝑿 and cos⁡𝒀.
Cosine o 𝑋= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐜𝐨𝐬 𝒙= 𝟏𝟓 𝟏𝟕 ≈𝟎.𝟖𝟖𝟐𝟒 𝟏𝟓 𝟖 𝟏𝟕 𝑿 𝒀 𝒁
Cosine of 𝑌= 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑌 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐜𝐨𝐬 𝒚= 𝟖 𝟏𝟕 ≈𝟎.𝟒𝟕𝟎𝟔

21. The Tangent Ratio
The last ratio is known as the Tangent Ratio.
Tangent of <𝐴= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐴 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 <𝐴 = 𝐵𝐶 𝐴𝐶
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪

22. The Tangent Ratio
The last ratio is known as the Tangent Ratio.
Tangent of <𝐵= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝐵 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝐵 = 𝐴𝐶 𝐵𝐶
Adjacent Leg
Opposite Leg
Hypotenuse 𝑨 𝑩 𝑪

23. Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁡𝑿 and 𝒕𝒂𝒏⁡𝒀.𝟓 𝟏𝟐 𝟏𝟑 𝒁 𝑿 𝒀

24. Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁡𝑿 and 𝒕𝒂𝒏⁡𝒀.𝟓 𝟏𝟐 𝟏𝟑 𝒁 𝑿 𝒀
Tangent of <𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋

25. Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁡𝑿 and 𝒕𝒂𝒏⁡𝒀.𝟓 𝟏𝟐 𝟏𝟑 𝒁 𝑿 𝒀
Tangent of <𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋
𝐭𝐚𝐧 𝑿= 𝟏𝟐 𝟓 =𝟐.𝟒

26. Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁡𝑿 and 𝒕𝒂𝒏⁡𝒀.𝟓 𝟏𝟐 𝟏𝟑 𝒁 𝑿 𝒀
Tangent of <𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋
𝐭𝐚𝐧 𝑿= 𝟏𝟐 𝟓 =𝟐.𝟒
Tangent of <𝑌= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑌 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑌

27. Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁡𝑿 and 𝒕𝒂𝒏⁡𝒀.𝟓 𝟏𝟐 𝟏𝟑 𝒁 𝑿 𝒀
Tangent of <𝑋= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑋 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑋
𝐭𝐚𝐧 𝑿= 𝟏𝟐 𝟓 =𝟐.𝟒
Tangent of <𝑌= 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 < 𝑌 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 𝑌
𝐭𝐚𝐧 𝒀= 𝟓 𝟏𝟐 ≈𝟎.𝟒𝟏𝟔𝟕

28. Putting it all together
In trigonometry, there is a saying that helps with memorizing how to set up the ratios of Sine, Cosine and Tangent.
See if you can get it from this:

29. Putting it all together
Sine: Opposite: Hypotenuse:

30. Putting it all together
Sine: Opposite: Hypotenuse:
Cosine: Adjacent: Hypotenuse:

31. Putting it all together
Sine: Opposite: Hypotenuse:
Cosine: Adjacent: Hypotenuse:
Tangent: Opposite: Adjacent:

32. Putting it all together
Sine: Opposite: Hypotenuse:
Cosine: Adjacent: Hypotenuse:
Tangent: Opposite: Adjacent:
SOH
CAH
TOA

33. Trig With Angles
Trig Ratios can also be used to find the values of specific angles.
For example, you can write tan (10) to represent the tangent of any angle of degree measure 10.
You can find these values by using either a calculator, or a table of values.

35. Trig With Angles
1.) tan 10° 4.) sin 45° 2.) sin 25° 5.) tan 60° 3.) cos 44° 6.) cos 30°

36. Trig With Angles
1.) tan 10° ≈𝟎.𝟏𝟕𝟔𝟑 4.) sin 45° ≈𝟎.𝟕𝟎𝟕𝟏 2.) sin 25° ≈𝟎.𝟒𝟐𝟐𝟔 5.) tan 60° ≈𝟏.𝟕𝟑𝟐𝟏 3.) cos 44° ≈𝟎.𝟕𝟏𝟗𝟑 6.) cos 30° ≈𝟎.𝟖𝟔𝟔𝟎

37. Examples: Find Missing Sides
Find the value of x. 𝟑𝟐 𝒙
𝟓𝟔°

38. Examples: Find Missing Sides
Find the value of x. 𝟑𝟐 𝒙
𝟓𝟔°
Solution:
tan 56° = 𝑥 32

39. Examples: Find Missing Sides
Find the value of x.
Solution:
tan 56° = 𝑥 32
𝑥=32∗ tan 56°
𝑥=32 ∗1.4826
𝒙=𝟒𝟕.𝟒𝟒𝟑𝟐 𝟑𝟐 𝒙
𝟓𝟔°

40. Examples: Find Missing Sides
Find the value of x. 𝟓𝟒 𝒙
𝟕𝟎°

41. Examples: Find Missing Sides
Find the value of x.
Solution:
sin 70° = 54 𝑥 𝟓𝟒 𝒙
𝟕𝟎°

42. Examples: Find Missing Sides
Find the value of x.
Solution:
sin 70° = 54 𝑥
𝑥∙ sin 70° =54
𝑥= 54 sin 70°
𝒙=𝟓𝟕.𝟒𝟔𝟓𝟔 𝟓𝟒 𝒙
𝟕𝟎°

43. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
67°
𝟏𝟐𝟎

44. Examples: Find Missing Sides
Find the values of x and y.
Solution (For x):
sin 67° = 𝑥 120 𝒚 𝒙
67°
𝟏𝟐𝟎

45. Examples: Find Missing Sides
Find the values of x and y.
Solution (For x):
sin 67° = 𝑥 120
𝑥=120∗ sin 67°
𝑥=120 ∗0.9205
𝒙=𝟏𝟏𝟎.𝟒𝟔 𝒚 𝒙
67°
𝟏𝟐𝟎

46. Examples: Find Missing Sides
Find the values of x and y.
Solution (For y):
cos 67° = 𝑦 120 𝒚 𝒙
67°
𝟏𝟐𝟎

47. Examples: Find Missing Sides
Find the values of x and y.
Solution (For y):
cos 67° = 𝑦 120
𝑦=120∗ cos 67°
𝑦=120 ∗0.3907
𝒚=𝟒𝟔.𝟖𝟖𝟒 𝒚 𝒙
67°
𝟏𝟐𝟎

48. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
𝟑𝟎° 𝟒𝟓

49. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
𝟑𝟎° 𝟒𝟓
Solution (For x):
tan 30° = 45 𝑥

50. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
𝟑𝟎° 𝟒𝟓
Solution (For x):
tan 30° = 45 𝑥
𝑥∗ tan 30°=45
𝑥= 45 tan 30°
𝒙=𝟕𝟕.𝟗𝟒𝟐𝟑

51. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
𝟑𝟎° 𝟒𝟓
Solution (For y):
sin 30° = 45 𝑦

52. Examples: Find Missing Sides
Find the values of x and y. 𝒚 𝒙
𝟑𝟎° 𝟒𝟓
Solution (For y):
sin 30° = 45 𝑦
𝑦∙ sin 30° =45
𝑦= 45 sin 30°
𝒚=𝟗𝟎

53. Final Check
Solve the value of x using trig ratios.
1.)

54. Final Check Solve the value of x using trig ratios. 1.) Solution:
𝑥≈15.004

55. Final Check
Solve the value of x using trig ratios.
2.)

56. Final Check Solve the value of x using trig ratios. 2.) Solution:
cos 63° = 23 𝑥
𝑥= 23 cos 63° 𝑥≈

57. Final Check
Solve the value of x using trig ratios.
3.)

58. Final Check Solve the value of x using trig ratios. Solution: 3.)
tan 38° = 20 𝑥
𝑥= 20 tan 38° 𝑥≈