1. A RATIO is a comparison of two numbers. For example;
Trigonometric Ratios
A RATIO is a comparison of two numbers. For example;
boys to girls
cats : dogs
right : wrong.
In Trigonometry, the comparison is between sides of a triangle ( right triangle).

2. CCSS: G.SRT.7
EXPLAIN and USE the relationship between the sine and cosine of complementary angles.

3. Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.  
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

4. Warm up Solve the equations: A) 0.875 = x/18 B) 24/y = .5
C) y/25 = .96

5. E.Q:
How can we find the sin, cosine, and the tangent of an acute angle?
How do we use trigonometric ratios to solve real-life problems?

6. Trig. Ratios Sine Cosine tangent Sin Cos Tan Sinθ = opposite side
Name
“say”
Sine
Cosine
tangent
Abbreviation
Abbrev.
Sin
Cos
Tan
Ratio of an angle measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
tanθ =opposite side
adjacent side

7. Three Trigonometric Ratios
Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.
Sine – abbreviated ‘sin’.
Ratio: sin θ = opposite side
hypotenuse
Cosine - abbreviated ‘cos’.
Ratio: cos θ = adjacent side
Tangent - abbreviated ‘tan’.
Ratio: tan θ = opposite side
adjacent side
Easy way to remember trig ratios:
SOH CAH TOA

8. Let’s practice… B c a C b A Write the ratio for sin A Sin A = o = a
h c
Write the ratio for cos A
Cos A = a = b
Write the ratio for tan A
Tan A = o = a
a b B c a
C b A
Let’s switch angles: Find the sin, cos and tan for Angle B:
Tan B = b a
Sin B = b c
Cos B = a c

9. Use these calculator keys
Make sure you have a calculator…
I want to find
Use these calculator keys
sin, cos or tan
ratio
SIN
COS
TAN
Angle measure
SIN-1
COS-1
TAN-1
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit

10. Let’s practice… C 2cm B 3cm A Process:
Find an angle that has a tangent (ratio) of 2 3
Round your answer to the nearest degree. C
2cm
B cm A
Process:
I want to find an ANGLE
I was given the sides (ratio)
Tangent is opp
adj
TAN-1(2/3) = 34°

11. Practice some more… Find tan A: 24.19 12 A 21 Find tan A: A
Tan A = opp/adj = 12/21
Tan A = .5714 8 4 A
Find tan A: 8
Tan A = 8/4 = 2

12. Trigonometric Ratios When do we use them?
On right triangles that are NOT or
Find: tan 45 1
Why?
tan = opp
hyp

13. Using trig ratios in equations
Remember back in 1st grade when you had to solve:
12 = x What did you do? 6
(6) (6)
72 = x
Remember back in 3rd grade when x was in the denominator?
12 = What did you do? x
(x) (x)
12x = 6
__ __
x = 1/2

14. In relation to the angle, what pieces do I have?
Ask yourself:
In relation to the angle, what pieces do I have?
x cm
15 cm
34°
Opposite and hypotenuse
Ask yourself:
What trig ratio uses Opposite and Hypotenuse?
SINE
Set up the equation and solve:
(15) (15)
Sin 34 = x 15
(15)Sin 34 = x
8.39 cm = x

15. In relation to the angle, what pieces do I have?
Ask yourself:
In relation to the angle, what pieces do I have?
53°
12 cm
Opposite and adjacent
Ask yourself:
What trig ratio uses Opposite and adjacent?
x cm
tangent
Set up the equation and solve:
(12) (12)
Tan 53 = x 12
(12)tan 53 = x
15.92 cm = x

16. In relation to the angle, what pieces do I have?
Ask yourself:
In relation to the angle, what pieces do I have?
x cm
Adjacent and hypotenuse
68°
Ask yourself:
What trig ratio uses adjacent and hypotnuse?
18 cm
cosine
Set up the equation and solve:
(x) (x)
Cos 68 = 18 x
(x)Cos 68 = 18
_____ _____
cos cos 68
X = 18
cos 68
X = cm

17. Remember to use the inverse function when you find theta
This time, you’re looking for theta.
Ask yourself:
In relation to the angle, what pieces
do I have?
42 cm
22 cm
Opposite and hypotenuse
THIS IS IMPORTANT!!
Ask yourself:
What trig ratio uses opposite and hypotenuse? θ
sine
Set up the equation (remember you’re looking for theta):
Sin θ = 22 42
Remember to use the inverse function when you find theta
Sin = θ 42
31.59°= θ

18. You’re still looking for theta.θ
THIS IS IMPORTANT!!
Ask yourself:
What trig ratio uses the parts I was given?
22 cm
17 cm
tangent
Set it up, solve it, tell me what you get.
tan θ = 17 22
tan = θ 22
37.69°= θ

19. Using trig ratios in equations
Remember back in 1st grade when you had to solve:
12 = x What did you do? 6
(6) (6)
72 = x
Remember back in 3rd grade when x was in the denominator?
12 = What did you do? x
(x) (x)
12x = 6
__ __
x = 1/2

20. Types of Angles
The angle that your line of sight makes with a line drawn horizontally.
Angle of Elevation
Angle of Depression

21. Indirect Measurement

22. SOA CAH TOA
SOA CAH TOA

23. Solving a right triangle
Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if the following one of the two situations exist:
Two side lengths
One side length and one acute angle measure

24. E.Q
How do we use right triangles to solve real life problems?

25. Note:
As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle.

26. WRITE THIS DOWN!!! In general, for an acute angle A:
If sin A = x, then sin-1 x = mA
If cos A = y, then cos-1 y = mA
If tan A = z, then tan-1 z = mA
The expression sin-1 x is read as “the inverse sine of x.”
On your calculator, this means you will be punching the 2nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle.

27. Example 1:
Solve the right triangle. Round the decimals to the nearest tenth.
HINT: Start by using the Pythagorean Theorem. You have side a and side b. You don’t have the hypotenuse which is side c—directly across from the right angle.

28. Example 1: (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem
Substitute values
c2 =
Simplify
c2 = 9 + 4
Simplify
c2 = 13
Find the positive square root
c = √13
Use a calculator to approximate
c ≈ 3.6

29. Example 1 continued Then use a calculator to find the measure of B:
2nd function
Tangent button 2
Divided by symbol
3 ≈ 33.7°

30. Finally Because A and B are complements, you can write
mA = 90° - mB ≈ 90° ° = 56.3°
The side lengths of the triangle are 2, 3 and √13, or about The triangle has one right angle and two acute angles whose measure are about 33.7° and 56.3°.

31. Ex. 2: Solving a Right Triangle (h)
Solve the right triangle. Round decimals to the nearest tenth.
25°
You are looking for opposite and hypotenuse which is the sin ratio.
sin H =
opp.
hyp.
Set up the correct ratio
13 sin 25° = h 13
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.4226) ≈ h
Use your calculator to approximate.
5.5 ≈ h

32. Ex. 2: Solving a Right Triangle (g)
Solve the right triangle. Round decimals to the nearest tenth.
25°
You are looking for adjacent and hypotenuse which is the cosine ratio.
cos G =
adj.
hyp.
Set up the correct ratio
13 cos 25° = g 13
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.9063) ≈ g
11.8 ≈ h
Use your calculator to approximate.

33. Using Right Triangles in Real Life
Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes.
A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

34. Solution:
Glide  = x°
15.7 miles
You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle.
59 miles
opp.
tan x° =
Use correct ratio
adj.
15.7
Substitute values
tan x° = 59
Key in calculator 2nd function,
tan 15.7/59 ≈ 14.9
 When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°.

35. B. Solution
Glide  = 19° h
When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth.
5 miles
opp.
tan 19° =
Use correct ratio
adj. h
Substitute values
tan 19° = 5 h 5
Isolate h by multiplying by 5.
5 tan 19° = 5
 The shuttle’s altitude is about 1.7 miles.
1.7 ≈ h
Approximate using calculator