Define the sine, cosine, and tangent ratios and their inverses. Find the measure of a missing side or angle.

1. Trigonometry
The student is able to (I can):
For any right triangle
• Define the sine, cosine, and tangent ratios and their
inverses
• Find the measure of a side given a side and an angle
• Find the measure of an angle given two sides
• Use trig ratios to solve problems

2. By the Angle-Angle Similarity Theorem, a right triangle with a
given acute angle is similar to every other right triangle with
the same acute angle measure. This means that the ratios
between the sides of those triangles are always the same.
Because these ratios are so useful, they were given names:
sinesinesinesine, cosinecosinecosinecosine, and tangenttangenttangenttangent. These ratios are used in the study
of trigonometrytrigonometrytrigonometrytrigonometry.

3. sine of ∠A
cosine of ∠A
tangent of ∠A
AAAA
hypotenuse
adjacent
opposite
∠
= =
leg opposite
sin
hypotenuse
A
A
leg adjacent to
cos
hypotenuse
A
A
∠
= =
leg opposite
tan
leg adjacent to
A
A
A
∠
= =
∠

4. Examples
I. Use the triangle to find the following ratios.
1. sin B = _____
2. cos B = _____
3. tan B = _____
A
B
C
8
15
17

5. Examples
I. Use the triangle to find the following ratios.
1. sin B = _____
2. cos B = _____
3. tan B = _____
A
B
C
8
15
17
8
17
15
17
8
15

6. To use the Nspire calculator to find tan 51°:
• From a New Document, press the µ key:
• Use the right arrow key (¢) to select tan and press ·:

7. • Type 5I and hit ·:
To use the calculator on your phone:
• Turn your phone landscape to access the scientific
calculator.
• Depending on your phone, you will either teither teither teither type the angle in
first and select tan, orororor select tan and then type in the
angle.

8. To find an angle, we use the inverseinverseinverseinverse trig functions (you will
sometimes hear them referred to as arcsine, arccosine, and
arctangent). On your calculator, these are listed as sin–1,
cos–1, and tan–1.
Ex. Find :
Press the µ button, and then the ¤ arrow to select sin–1.
Then enter 8p17·. You should get 28.07…
This means that the angle opposite a leg of 8 with a
hypotenuse of 17 will measure around 28˚.
1 8
sin
17
−  
 
 

9. You will be expected to memorize these ratio relationships.
There are many hints out there to help you keep them
straight. The most common is SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA , where
A mnemonic I like is “Some Old Hippie Caught Another
Hippie Trippin’ On Acid.”
Or “Silly Old Hitler Couldn’t Advance His Troops Over Africa.”
pp
in
yp
O
S
H
=
dj
os
yp
A
C
H
=
pp
an
dj
O
T
A
=

10. We can use the trig ratios to find either missing sides or
missing angles of right triangles. To do this, we set up
equations and solve for the missing part.
When setting up your equation, use the following
pattern:
where “trig” is sine, cosine, or tangent ratio, A is the
angle, and the two sides are the ones that go with that
ratio. Now solve for the missing piece.
( )° =
side
trig
side
A

11. II. Find the lengths of the sides to the nearest tenth.
1.
2.
x (opp)
15
(adj)
58°
26
(hyp)
x
(adj)
46°
( )
( )
° =
= °
≈
tan 58
15
15tan 58
24.0
x
x
( )
( )
° =
= °
≈
cos 46
26
26cos 46
18.1
x
x
(x on top: multiply)

12. 3.
Also, notice that the sides are multiplying/dividing the trig
ratio, notnotnotnot the angle.
x
(hyp) 16
(opp)
37°
( )
( )
( )
° =
° =
=
°
≈
16
sin 37
sin 37 16
16
sin 37
26.6
x
x
x
(x on bottom: divide)

13. III. Find the missing angle to the nearest whole degree.
26 (hyp)
19 (opp)
x°
1
19
sin
26
19
sin
26
47
x
x −
° =
 
=  
 
≈ °
You’ll learn more about
this in Algebra II, but a
function and its inverse
cancel each other out.
So
( )−
° = °1
sin sinx x