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13.1 Right Triangle Trigonometry
©2002 by R. Villar
All Rights Reserved
Right Triangle Trigonometry
Let’s consider a right triangle, one of whose acute angles is

The three sides of the triangle are the hypotenuse, the side
opposite  , and the side adjacent to  .
Ratios of a right triangle’s three sides are
used to define six trigonometric functions:
hypotenuse
sine, cosine, tangent,
cosecant, secant, and cotangent
opposite

Right Triangle Definition of Trig Functions
Let  be an acute angle of a right triangle.
sin  = opp
hyp
cos  = adj
hyp
tan  = opp
adj
csc  = hyp
opp
sec  = hyp
adj
cot  = adj
opp
adjacent
Example:
Evaluate the six trigonometric functions of the
angle  shown.
Use the Pythagorean Theorem to find the
length of the adjacent side…
a2 + 122 = 132
a2 = 25
a=5
adj = 5
opp = 12
hyp = 13
12
13

sin  = opp
hyp
=
12
13
cos  = adj = 5
13
hyp
tan  = opp = 5
12
adj
csc  = hyp
opp
=
13
12
sec  = hyp =
adj
cot  = adj =
opp
13
5
12
5
Example: Given cos  = 4/5 , find cot  .
Since cos  = adj = 4 , the triangle looks like this…
hyp
5
5

Use the Pythagorean Theorem to find the
length of the opposite side…
a2 + 42 = 52
a2 = 9
a=3
adj = 4
opp = 3
hyp = 5
cot

= adj =
opp
4
3
4
The angles 30º , 45º , and 60º occur frequently in trigonometry.
Use the triangle to find the trig. functions when
 = 30º
adj =
3
opp = 1
hyp = 2
30º
2
3
1
sin 30º = opp =
hyp
1
2
cos 30º = adj  3
2
hyp
tan 30º = opp  1  3
3
3
adj
csc 30º = hyp =
opp
2
1
2 3
sec 30º = hyp  2

3
3
adj
cot 30º = Remember,
adj  3 the radical
in
the denominator
1
opp
must be simplified.
Notice that the same triangle can be used when the acute angle is 60º :
Use the triangle to find the trig. functions when
 = 60º
adj = 1
opp = 3
hyp = 2
2
3
60º
1
sin 60º = opp  3
2
hyp
cos 60º = adj =
hyp
csc 60º = hyp  2 2 3

3
3
opp
sec 60º = hyp =
adj
tan 60º = opp  3
1
adj
1
2
2
1
1
cot 60º = adj
3


opp
3
3
What if the acute angle is 45º ?
Use the triangle to find the trig. functions when
 = 45º
adj = 1
opp = 1
hyp =
2
2
1
45º
1
sin 45º = opp  1  2
2
2
hyp
2
1
cos 45º = adj


2
hyp
2
tan 45º = opp  1  1
adj 1
2
csc 45º = hyp

1
opp
sec 45º = hyp  2
1
adj
cot 45º = adj  1  1
opp 1
You will need to remember how to re-create these triangles.