7-5 and 7-6: Apply Trigonometric Ratios

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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 π‘ͺ Example: For βˆ†π‘¨π‘©π‘ͺ
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 π‘ͺ Hypotenuse Opposite Leg Adjacent Leg

6 Trigonometry 𝑩 𝑨 π‘ͺ Example: For βˆ†π‘¨π‘©π‘ͺ
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 π‘ͺ For Example: 𝑩π‘ͺ is the Leg Opposite to <𝑨 And 𝑨π‘ͺ is the Leg Adjacent to <𝑨 Hypotenuse Opposite Leg Adjacent Leg

7 The Sine Ratio Sine of <𝐴= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐴 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐡𝐢 𝐴𝐡
The first ratio is known as the Sine Ratio. Sine of <𝐴= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐴 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐡𝐢 𝐴𝐡 Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ

8 The Sine Ratio Sine of <𝐡= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐡 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐴𝐢 𝐴𝐡
The first ratio is known as the Sine Ratio. Sine of <𝐡= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐡 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐴𝐢 𝐴𝐡 Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ

9 Sine Examples From the given triangle, find π’”π’Šπ’β‘π‘Ώ and π’”π’Šπ’β‘π’€. πŸπŸ“ πŸ– πŸπŸ• 𝑿
𝒁

10 Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
Sine Examples From the given triangle, find π’”π’Šπ’β‘π‘Ώ and π’”π’Šπ’β‘π’€. Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁

11 Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
Sine Examples From the given triangle, find π’”π’Šπ’β‘π‘Ώ and π’”π’Šπ’β‘π’€. Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐬𝐒𝐧 𝒙 = πŸ– πŸπŸ• β‰ˆπŸŽ.πŸ’πŸ•πŸŽπŸ” πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁

12 Sine Examples From the given triangle, find π’”π’Šπ’β‘π‘Ώ and π’”π’Šπ’β‘π’€.
Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐬𝐒𝐧 𝒙 = πŸ– πŸπŸ• β‰ˆπŸŽ.πŸ’πŸ•πŸŽπŸ” πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁 Sine of π‘Œ= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < π‘Œ β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’

13 Sine Examples From the given triangle, find π’”π’Šπ’β‘π‘Ώ and π’”π’Šπ’β‘π’€.
Sine of 𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐬𝐒𝐧 𝒙 = πŸ– πŸπŸ• β‰ˆπŸŽ.πŸ’πŸ•πŸŽπŸ” πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁 Sine of π‘Œ= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < π‘Œ β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐬𝐒𝐧 π’š= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’

14 The Cosine Ratio The second ratio is known as the Cosine Ratio. Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ Cosine of <𝐴= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝐴 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐴𝐢 𝐴𝐡

15 The Cosine Ratio The second ratio is known as the Cosine Ratio. Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ Cosine of <𝐡= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝐡 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ = 𝐡𝐢 𝐴𝐡

16 Cosine Examples From the given triangle, find cos⁑𝑿 and cos⁑𝒀. 𝒀 πŸπŸ• πŸ–
πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁

17 Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
Cosine Examples From the given triangle, find cos⁑𝑿 and cos⁑𝒀. Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁

18 Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
Cosine Examples From the given triangle, find cos⁑𝑿 and cos⁑𝒀. Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐜𝐨𝐬 𝒙= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’ πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁

19 Cosine Examples 𝐜𝐨𝐬 𝒙= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’
From the given triangle, find cos⁑𝑿 and cos⁑𝒀. Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐜𝐨𝐬 𝒙= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’ πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁 Cosine of π‘Œ= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < π‘Œ β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’

20 Cosine Examples 𝐜𝐨𝐬 𝒙= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’ 𝐜𝐨𝐬 π’š= πŸ– πŸπŸ• β‰ˆπŸŽ.πŸ’πŸ•πŸŽπŸ”
From the given triangle, find cos⁑𝑿 and cos⁑𝒀. Cosine o 𝑋= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐜𝐨𝐬 𝒙= πŸπŸ“ πŸπŸ• β‰ˆπŸŽ.πŸ–πŸ–πŸπŸ’ πŸπŸ“ πŸ– πŸπŸ• 𝑿 𝒀 𝒁 Cosine of π‘Œ= 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < π‘Œ β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ 𝐜𝐨𝐬 π’š= πŸ– πŸπŸ• β‰ˆπŸŽ.πŸ’πŸ•πŸŽπŸ”

21 The Tangent Ratio The last ratio is known as the Tangent Ratio. Tangent of <𝐴= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐴 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ <𝐴 = 𝐡𝐢 𝐴𝐢 Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ

22 The Tangent Ratio The last ratio is known as the Tangent Ratio. Tangent of <𝐡= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝐡 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝐡 = 𝐴𝐢 𝐡𝐢 Adjacent Leg Opposite Leg Hypotenuse 𝑨 𝑩 π‘ͺ

23 Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁑𝑿 and 𝒕𝒂𝒏⁑𝒀.
πŸ“ 𝟏𝟐 πŸπŸ‘ 𝒁 𝑿 𝒀

24 Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁑𝑿 and 𝒕𝒂𝒏⁑𝒀.
πŸ“ 𝟏𝟐 πŸπŸ‘ 𝒁 𝑿 𝒀 Tangent of <𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋

25 Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁑𝑿 and 𝒕𝒂𝒏⁑𝒀.
πŸ“ 𝟏𝟐 πŸπŸ‘ 𝒁 𝑿 𝒀 Tangent of <𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 𝐭𝐚𝐧 𝑿= 𝟏𝟐 πŸ“ =𝟐.πŸ’

26 Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁑𝑿 and 𝒕𝒂𝒏⁑𝒀.
πŸ“ 𝟏𝟐 πŸπŸ‘ 𝒁 𝑿 𝒀 Tangent of <𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 𝐭𝐚𝐧 𝑿= 𝟏𝟐 πŸ“ =𝟐.πŸ’ Tangent of <π‘Œ= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < π‘Œ 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < π‘Œ

27 Tangent Examples From the given right triangle, find 𝒕𝒂𝒏⁑𝑿 and 𝒕𝒂𝒏⁑𝒀.
πŸ“ 𝟏𝟐 πŸπŸ‘ 𝒁 𝑿 𝒀 Tangent of <𝑋= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < 𝑋 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < 𝑋 𝐭𝐚𝐧 𝑿= 𝟏𝟐 πŸ“ =𝟐.πŸ’ Tangent of <π‘Œ= 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ < π‘Œ 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ < π‘Œ 𝐭𝐚𝐧 𝒀= πŸ“ 𝟏𝟐 β‰ˆπŸŽ.πŸ’πŸπŸ”πŸ•

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.

34

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

36 Trig With Angles 1.) tan 10Β° β‰ˆπŸŽ.πŸπŸ•πŸ”πŸ‘ 4.) sin 45Β° β‰ˆπŸŽ.πŸ•πŸŽπŸ•πŸ 2.) sin 25Β° β‰ˆπŸŽ.πŸ’πŸπŸπŸ” 5.) tan 60Β° β‰ˆπŸ.πŸ•πŸ‘πŸπŸ 3.) cos 44Β° β‰ˆπŸŽ.πŸ•πŸπŸ—πŸ‘ 6.) cos 30Β° β‰ˆπŸŽ.πŸ–πŸ”πŸ”πŸŽ

37 Examples: Find Missing Sides
Find the value of x. πŸ‘πŸ 𝒙 πŸ“πŸ”Β°

38 Examples: Find Missing Sides
Find the value of x. πŸ‘πŸ 𝒙 πŸ“πŸ”Β° Solution: tan 56Β° = π‘₯ 32

39 Examples: Find Missing Sides
Find the value of x. Solution: tan 56Β° = π‘₯ 32 π‘₯=32βˆ— tan 56Β° π‘₯=32 βˆ—1.4826 𝒙=πŸ’πŸ•.πŸ’πŸ’πŸ‘πŸ πŸ‘πŸ 𝒙 πŸ“πŸ”Β°

40 Examples: Find Missing Sides
Find the value of x. πŸ“πŸ’ 𝒙 πŸ•πŸŽΒ°

41 Examples: Find Missing Sides
Find the value of x. Solution: sin 70Β° = 54 π‘₯ πŸ“πŸ’ 𝒙 πŸ•πŸŽΒ°

42 Examples: Find Missing Sides
Find the value of x. Solution: sin 70Β° = 54 π‘₯ π‘₯βˆ™ sin 70Β° =54 π‘₯= 54 sin 70Β° 𝒙=πŸ“πŸ•.πŸ’πŸ”πŸ“πŸ” πŸ“πŸ’ 𝒙 πŸ•πŸŽΒ°

43 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 67Β° 𝟏𝟐𝟎

44 Examples: Find Missing Sides
Find the values of x and y. Solution (For x): sin 67Β° = π‘₯ 120 π’š 𝒙 67Β° 𝟏𝟐𝟎

45 Examples: Find Missing Sides
Find the values of x and y. Solution (For x): sin 67Β° = π‘₯ 120 π‘₯=120βˆ— sin 67Β° π‘₯=120 βˆ—0.9205 𝒙=𝟏𝟏𝟎.πŸ’πŸ” π’š 𝒙 67Β° 𝟏𝟐𝟎

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

47 Examples: Find Missing Sides
Find the values of x and y. Solution (For y): cos 67Β° = 𝑦 120 𝑦=120βˆ— cos 67Β° 𝑦=120 βˆ—0.3907 π’š=πŸ’πŸ”.πŸ–πŸ–πŸ’ π’š 𝒙 67Β° 𝟏𝟐𝟎

48 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 πŸ‘πŸŽΒ° πŸ’πŸ“

49 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 πŸ‘πŸŽΒ° πŸ’πŸ“ Solution (For x): tan 30Β° = 45 π‘₯

50 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 πŸ‘πŸŽΒ° πŸ’πŸ“ Solution (For x): tan 30Β° = 45 π‘₯ π‘₯βˆ— tan 30Β°=45 π‘₯= 45 tan 30Β° 𝒙=πŸ•πŸ•.πŸ—πŸ’πŸπŸ‘

51 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 πŸ‘πŸŽΒ° πŸ’πŸ“ Solution (For y): sin 30Β° = 45 𝑦

52 Examples: Find Missing Sides
Find the values of x and y. π’š 𝒙 πŸ‘πŸŽΒ° πŸ’πŸ“ Solution (For y): sin 30Β° = 45 𝑦 π‘¦βˆ™ sin 30Β° =45 𝑦= 45 sin 30Β° π’š=πŸ—πŸŽ

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:
π‘₯β‰ˆ15.004

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 63Β° = 23 π‘₯ π‘₯= 23 cos 63Β° π‘₯β‰ˆ

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 38Β° = 20 π‘₯ π‘₯= 20 tan 38Β° π‘₯β‰ˆ

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