Geometry
The three primary Trigonometric Ratios are
Sine, Cosine, and Tangent.
As we learned previously, triangles with the same angle measures have
proportional sides.
If you know one angle in a right triangle, you can use
sin, cos, and tan to find the ratio of the lengths of its sides.
Sine, Cosine, and Tangent
Sin x
o
=
Cos x
o
=
Tan x
o
=
12.1
Trigonometry
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
x
o
adjacent
opposite
hypotenuse
Find sin x, cos x, and tan x in the right triangles below:
1.
2.
Sin x
o
=
Cos x
o
=
Tan x
o
=
x
o
5cm
12cm
13cm
5cm
3cm
4cm
x
o
If you know an angle, you can use a calculator to find the ratios:
1.
sin 34
o
=
2.
cos 55
o
=
3.
tan 18
o
=
Sin x
o
=
Cos x
o
=
Tan x
o
=
Use a calculator to find the missing lengths below:
1.
2.
5cm
x
y
27
o
58
o
22cm
b
a

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Geometry 12.1
Trigonometry
You can use trigonometric inverses to find an angle based on a ratio:
Sin, Cos, and Tan
turn an angle into a ratio.
Sin
-1
, Cos
-1
, and Tan
-1
turn a ratio into an angle measure.
Example:
Find the measure of angle x in each triangle below:
(make sure your calculator is set to degrees not radians)
1.
2.
5cm
12cm
13cm
x
o
x
o
109 in
60 in
91 in
Practice:
Find the missing information in each right triangle below:
(round to the hundredth)
1.
2.
3.
x
o
15cm
11cm
x
5cm
35
o
x
9cm
15
o
Practice:
Find the apothem in each regular polygon below:
1.
2.
3.
Find the sides.
8cm
10cm
6cm