1. O B J E C T I V E S
S T U D E N T S W I L L E X P L O R E T H E C O N N E C T I O N B E T W E E N S I M I L A R
R I G H T T R I A N G L E S A N D T R I G O N O M E T R I C R A T I O S
S T U D E N T S W I L L F O R M A C O N J E C T U R E A B O U T T H E
R E L A T I O N S H I P B E T W E E N S I N E A N D C O S I N E O F
C O M P L E M E N T A R Y A N G L E S I N A R I G H T T R I A N G L E
S T U D E N T S W I L L A P P L Y S I N E , C O S I N E , A N D T A N G E N T R A T I O S T O
S O L V E R I G H T T R I A N G L E S
Similar Triangles and Trigonometric
Ratios

2. Review of Proving Triangles Similar
 Three ways to prove triangles similar
 SSS
 If all three sides on two triangles are proportional, then the
triangles are similar
 SAS
 If two sides are proportional and the included angles are
congruent, then the triangles are similar
 AA
 If two angles are congruent, then the two triangles are similar

4. Terms to know for Trigonometry
θ
Reference Angle
Opposite side
(o)
Adjacent Side
(a)
Hypotenuse
(h)

5. θ
Reference Angle
Opposite side
(o)
Adjacent Side
(a)
Hypotenuse
(h)
Trigonometric Ratios
Sine
Sin θ = o
h
Cosine
Cos θ = a
h
Tangent
Tan θ = o
a

6. Ways to Memorize
Tangent
Tan θ = o
a
Sine
Sin θ = o
h
Cosine
Cos θ = a
h
Ohio Has A Heroic Offense Again
Soh Cah Toa

7. Writing the Trigonometric Ratios
x
5
12
13
(o)
(a)
(h)

8. Now lets look at the other acute angle
θ
5
12
13
(a)
(h)
(o)
What do you notice about the sine, cosine, and tangent
from the two acute angles in the same right triangle?
x

9. Using Trigonometry Ratios
Step 1: Identify which trigonometric
ratio we are using based on the
sides given
38°
15
xO
h=sin
Sin =
Step 2: Plug the numbers into the ratio

10. Solve
sin 38°=
.6157 =
.6157 x = 15
x= 24.36
15
x
38°
x
15
15
x
NOTE** Sin 38°=.6157
Now, what if we want to find the missing side
PYTHAGOREAN THEOREM!!!

11. Inverse Functions
What if we need to find the angle?
Step 1: Identify which trigonometric
ratio we are using based on the
sides given
Step 2: Plug the numbers into the ratio
Tan =
x
8
16
O
a
=tan

12. Solving Inverse Functions
Tan =
To solve, switch the x and the , making the tangent
its inverse
Tan-1 =
Now use your Calculators:
26.57 °=x
8
16
8
16
x

13. Application
Pythagoras is walking up the stairs. For fun, he decides
to do a little math along the way. Before walking up
the stairs, he measures that the bottom of the stairs
make a 50° angle with the ground. If the stairs reach
6.3 meters on the wall, can you tell Pythagoras the
length of the stairs.
** Try on your Own

14. Answer
sin 5o° =
.76604 =
.76604 x = 6.3
x= 8.22 meters
50°
6.3
x
x
6.3
x
6.3

15. What did we learn today?
 The three trigonometric ratios, sine, cosine, and
tangent
 How to use the trigonometric ratios to solve for
missing sides and angles
 Practical applications of their uses

16. Exit Slip
 Find the missing angles and sides
27
53°