1. Unit 8 – Right Triangle Trig Trigonometric Ratios in Right Triangles

2. Trigonometric Ratios are based on the Concept of Similar Triangles!

3. All 45º- 45º- 90º Triangles are Similar!
45 º 1
45 º 2
45 º 1

4. All 30º- 60º- 90º Triangles are Similar!2
60º
30º 1
60º
30º 4
60º
30º 2 1

5. Trigonometric functions -- the ratios of sides of a right triangle.
Similar Triangles Always Have the Same Trig Ratio Answers!
adjacent
opposite
hypotenuse c b
leg
SINE
COSINE 
TANGENT
leg a
They are abbreviated using their first 3 letters

6. Oh, I'm acute!
This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.  5 4
So am I!  3

7. INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE
Here is a mnemonic to help you memorize the ratios.  c b
opposite
SOHCAHTOA
SOHCAHTOA 
adjacent
INE a
OSINE
ANGENT
PPOSITE
DJACENT
YPOTENUSE
PPOSITE
DJACENT
YPOTENUSE

8. Let's try finding some trig functions with some numbers.
It is important to note WHICH angle you are talking about when you find the value of the trig function. 
Let's try finding some trig functions with some numbers.
hypotenuse 5 c b 4
opposite
opposite 
adjacent a 3
sin  =
Use a mnemonic and figure out which sides of the triangle you need for tangent.
Use a mnemonic and figure out which sides of the triangle you need for sine.
tan  =

9. How do the trig answers for  and  relate to each other?
hypotenuse c 5 b 4
opposite
opposite 
adjacent a 3

10. 10.8 9 A 6 Find the sine, the cosine, and the tangent of angle A.
Give a fraction and decimal answer (round to 4 places).
10.8 9 A 6
Now, figure out your ratios.

11. Tangent 14º 0.25 The Tangent of an angle is the ratio of the
opposite side of a triangle to its adjacent side.
hypotenuse
1.9 cm
opposite
adjacent
14º
7.7 cm
Tangent 14º
0.25

12. Tangent A =
3.2 cm
24º
7.2 cm
Tangent 24º
0.45

13. As an acute angle of a triangle approaches 90º, its tangent
Tangent A =
As an acute angle of a triangle
approaches 90º, its tangent
becomes infinitely large
very
large
Tan 89.9º = 573
Tan 89.99º = 5,730
etc.
very small

14. Sine A = Cosine A = Since the sine and cosine functions always
have the hypotenuse as the denominator,
and since the hypotenuse is the longest side,
these two functions will always be less than 1.
Sine A =
Cosine A =
Sine 89º = .9998 A
Sine 89.9º =

15. Sin α =
7.9 cm
3.2 cm
24º
Sin 24º
0.41

16. Cosine β =
7.9 cm
46º
5.5 cm
Cos 46º
0.70

17. Ex. Solve for a missing value using a trig function.
tan 20 55 )
Now, figure out which trig ratio you have and set up the problem.

18. Ex: 2 Find the missing side. Round to the nearest tenth.
80 ft x
tan 80  ( 72 ) ) =
Now, figure out which trig ratio you have and set up the problem.

19. Ex: 3 Find the missing side. Round to the nearest tenth.
Now, figure out which trig ratio you have and set up the problem.

20. Ex: 4 Find the missing side. Round to the nearest tenth.
20 ft x

21. Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.)

22. Ex. 1: Find . Round to four decimal places.
tan
17.2 9 )
17.2 9
Now, figure out which trig ratio you have and set up the problem.
Make sure you are in degree mode (not radians).

23. Ex. 2: Find . Round to three decimal places.7
2nd
cos 7 23 ) 23
Make sure you are in degree mode (not radians).

24. Ex. 3: Find . Round to three decimal places.
200
400
2nd
sin
200
400 )
Make sure you are in degree mode (not radians).

25. When we are trying to find a side we use sin, cos, or tan.
When we are trying to find an angle
we use sin-1, cos-1, or tan-1.

26. 4 x A plane takes off from an airport an an angle of 18º and
a speed of 240 mph. Continuing at this speed and angle,
what is the altitude of the plane after 1 minute?
After 60 sec., at 240 mph, the plane
has traveled 4 miles 4 x
18º

27. SohCahToa Soh Sine A = Sine 18 = 0.3090 = x = 1.236 miles or= 1
x = miles or
6,526 feet 4 x
opposite
hypotenuse
18º

28. An explorer is standing 14.3 miles from the base of
Mount Everest below its highest peak. His angle of
elevation to the peak is 21º. What is the number of feet
from the base of Mount Everest to its peak?
Tan 21 = = 1
x = 5.49 miles
= 29,000 feet x
21º
14.3

29. A swimmer sees the top of a lighthouse on the
edge of shore at an 18º angle. The lighthouse is
150 feet high. What is the number of feet from the
swimmer to the shore?
0.3249x = 150
Tan 18 =
0.3249
0.3249 = 1
X = ft
150
18º x

30. A dragon sits atop a castle 60 feet high. An archer
stands 120 feet from the point on the ground directly
below the dragon. At what angle does the archer
need to aim his arrow to slay the dragon?
Tan x =
Tan x =
0.5
Tan-1(0.5) =
26.6º 60 x
120

31. Solving a Problem with the Tangent Ratio
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
h = ? 1 2
60º
53 ft
Why?

32. Ex.
A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?
tan 71.5° ?
tan 71.5°
71.5°
y = 50 (tan 71.5°) 50
y = 50 ( )

33. Ex. 5
A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge?
cos 60°
x (cos 60°) = 200
200
60° x x
X = 400 yards

34. Trigonometric Functions on a Rectangular Coordinate Systemx y q
Pick a point on the
terminal ray and drop a perpendicular to the x-axis. r y x
The adjacent side is x
The opposite side is y
The hypotenuse is labeled r
This is called a
REFERENCE TRIANGLE.

35. Trigonometric Ratios may be found by:
45 º 1
Using ratios of special triangles
For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)