Trigonometria

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3. The opposite leg of a right triangle is the leg which does not touch the vertex
of the angle that is named in the trig function
x
C
4. The adjacent leg of a right triangle is the leg A
which does touch the vertex of the angle that is
named in the trig function; for example:
When evaluating the trig functions for angle A
in this right triangle, legy is the opposite leg
y
z
for angle A because it does not touch point A;
however, leg x is the adjacent leg for angle A
because it does touch point A; the hypotenuse
B
is sidez
In the same right triangle, leg x is the opposite
leg for angle B because it does not touch point B; however, leg y is the adjacent leg for angle B because it does touch point B
(NOTICE: The leg that is the opposite legfor angle A is the same leg that is the adjacent leg for angle B, and the leg that is the adjacent leg for angle A is the same leg
that is the opposite leg for angle B; the hypotenuse is never considered as the
opposite side nor as the adjacent side because it is not a leg)
5. Since trig functions are ratios, and ratios can be written as decimal numbers,
trig functions are either converted todecimal numbers or left as radical
expressions in lowest terms (for example, 3 or .866); for example, in the right
2
triangles above, if:

Trigonometry (trig) means measurement of triangles; it is usually studied as measurements of sides and angles of triangles and as points on a unit circle; this study guide
is basically separated into these two main sections: trig with triangles and trig with a
unitcircle; in trigonometry, the measures of angles are usually represented by letters
from the Greek alphabet; the Greek letters θ, α, ν, and β will be used throughout this
study guide to represent angle measures

TRIG WITH TRIANGLES

a=

c=hypote

leg
=

9

A. Right Triangle
1. A right triangle is a triangle with
exactly one 90º (right) angle
2. The hypotenuse is the longest side
of a right triangle,and is always
located opposite the right (90º) angle
3. The two shorter sides of a right triangle are both called legs
4. The Pythagorean Theorem (leg2+ leg2 = hypotenuse2 or a2 + b2 = c2 where a
and b are leg lengths and c is the hypotenuse length) may be used to find the
length of any third side of a right triangle when any two side lengths are known
a. When the two leg lengths
a2 + b2 = c2
areknown, square the
2 + 162 = c2
length of each leg, add
9
these two squares together
81 + 256 = c2
and square root the result337 = c2
ing sum; for example:
b=

nuse

leg

=1

√337 = c
18.36 = c = hypotenuse

6
a=leg=

8

10

b=leg

se=

nu

ote

hyp
c=

b. When the length of the
hypotenuse and either leg
are known, square the
length of the hypotenuse,
square the length of the leg,
subtract thesetwo squares,
and square root the resulting difference; for example:

z = 9, y = 7, and x = √32 = 4√2 then
y
7=
sin A = cos B = z =
9
x = 4√2 =
cos A = sin B = z
9
y=7 =
tan A = x
4√2
x = 4√2 =
tan B = y
7

a2 + b 2 = c2
82 + b2 = 102
64 + b2 = 100
b2 = 100 - 64
b2 = 36
b = √36
leg = b = 6

.6285
1.2374
.8081

6. Using the trig function decimal number values to find or use angle measures
requireseither a trig function chart or a calculator with trig function options;
for example, if you have found that sin α = .7778 then, by using either a trig
chart or calculator, the measure of angle α is about 51º

t degrees
•
•
•
51º 00'
10
20
30
40
50

5. Special right triangles exist that are used so often that the relationships of the
side lengths should be memorized
a. 30º-60º-90º
triangles
haveside lengths with
c = hypotenuse = 2x
ratios of 1: 3 :2; that is,
b = longest leg = x√3
the longest leg is always
a = shortest leg = x
30º
3 times the length of the
b
shortest leg, and the
c If a = 5 then c = 2 • 5 = 10
hypotenuse is always 2
and b = 5 • √3 = 8.7
times the shortest leg.
If b = 8 then a = 8 ÷ √3 = 4.6
This relationship can be
60º
and c = 2 • 8 ÷ √3 = 9.2
used to find all side
a...