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

1. 4.11.4 Trigonometry
The student is able to (I can):
For any right triangle
• Define the sine, cosine, and tangent ratios and their• 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, andwere given names: sinesinesinesine, cosinecosinecosinecosine, and
tangenttangenttangenttangent. These ratios are used in the
study of trigonometry.

3. sine sine of ∠A
AAAA
hypotenuse
adjacent
opposite
leg opposite A
sinA
hypotenuse
∠
= =
cosine
tangent
cosine of ∠A
tangent of ∠A
hypotenuse
leg adjacent to A
cosA
hypotenuse
∠
= =
leg opposite A
tanA
leg adjacent to A
∠
= =
∠

4. We can use the trig ratios to find either
missing sides or missing angles of right
triangles. To do this, we will set up
equations and solve for the missing part.
In order to figure out the sine, cosine, and
tangent ratios, we can use either a
calculator or a trig table.

5. 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• Use the right arrow key (¢) to select tan
and press ·:

6. • Type 5I and hit ·:
To use the calculator on your phone:
• Turn your phone landscape to access
the scientific calculator.
• Type the angle in first, thenthenthenthen select tan.

7. To use a trig table to find cos 52°:
• Locate 52° on the table.
• Scan over to the Cos column and find
the value.
• cos 52° = .6157

8. To find an angle, we use the inverseinverseinverseinverse trig
functions (in more advanced classes, you
will 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 :−  
 
 
1 8
sin
17
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˚.
 
 17

9. To find an angle using a trig table, just find
the appropriate trig column, find the
closest value, and read back to the angle.
Ex. Find ( )−1
tan 0.35
0.35 is closer to 0.3443 than it is to
0.3640, so our answer would be 19˚.

10. 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
pO
S
p
in
pHy
=
dA
C
j
os
pHy
=
pO
T
p
an
jAd
=
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.”

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

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

13. Examples I. Use the triangle to find the following
ratios.
4. sin B = _____
A
B
C
8
15
17
8
174. sin B = _____
5. cos B = _____
6. tan B = _____
17
15
17
8
15

14. Examples II. Find the lengths of the sides to the
nearest tenth.
1.
x (opp)
15
(adj)
58°
° =
= °
≈
x
tan58
15
x 15tan58
24.0
2. 26
(hyp)
x
(adj)
46°
° =
= °
≈
x
cos46
26
x 26cos46
18.1

15. III. Find the missing angle to the nearest
whole degree.
26 (hyp)
19 (opp)
xº
° =
19
sinx
26
−  
=  
 
1 19
x sin
26
≈ °x 47