Download Trigonometry

Introduction to Trigonometric Ratios
Trigonometry is the study of triangles, most commonly, right triangles.
Since similar triangles have the same angles and proportional sides, then the following
proportion statements are all true:
A
H
A
H
a
h
x
90º
90º
x
90º
B
x
b
B
A/H = A/H = a/h
B/H = B/H = b/h
A/B = A/B = a/b
These ratios remain the same for any right triangle with an acute angle of xº regardless of
the size of the triangle.
So we give them each a name. Sine (xº)
Cosine (xº)
and
Tangent (xº)
These are just names of functions that refer to the ratios above:
Sine (xº) = A/H
Cosine (xº) = B/H
Tangent (xº) = A/B
This is true even if the triangles are rotated. A is always across from x, B is always next
to X, and H is always the longest side (the hypotenuse).
A
x
B
A
b
h
H
H
x
a
x
B
3
x
4
5
sin(x) = 4/5 = .8000
cos(x) = 3/5 = .6000
tan(x) = 4/3 = 1.3333
Trigonometry cont.
“soh, cah, toa”
sine X = opposite leg
hypotenuse
cosine X = adjacent leg
hypotenuse
hypotenuse
tangent X = opposite leg =
adjacent leg
opposite
leg
angle X
adjacent leg
If you know the angle measure of “x” and any one side, you can find the other sides
using a trig ratio.
?
sin(60º) = ?/10
.8660 = ?/10
8.66 = ?
10
60º
cos(52º) = 3/?
?*cos(52º) = 3
? = 3/cos(52º) = 4.87
3
52º
?
Try these:
7
?
8
49º
61º
?
First, decide which ratio you need. Then write an equation involving the information you
know. On the calculator, push the sin/cos/tan button and then type the angle and hit enter.
Be sure you are in degree mode. Put this value into your ratio equation and solve for the
unknown side length.
Finding The Angle If You Know at Least Two Sides
If you know two of the sides and want to find the angle, you need to use the inverse
functions (sin-1/cos-1/tan-1). On the calculator, press the 2nd key and then the trig function
key needed. Enter the ratio of the two sides and hit enter.
Ratio:
Angle (use the inverse functions)
tanX = 1
then X =tan-1(1) = or 2nd tan (1) on the calculator = 45º
sinX = 0.8660
X= sin-1(0.8660) = 60º
cosX = 0.7986
X= cos-1(0.7986) = 37º
tanX = 5/7
X= tan-1(5/7) = 35.5º
So you can use this in a triangle:
sin ( x) = 12/13
so: sin-1(12/13) = x = 67.4º
12
3
13
xº
xº
tan(x) = 3/4
so: tan -1(3/4) = x
x = 36.9º
4
Don’t forget you can use the Pythagorean Theorem to find the third side!
Try to find the missing values:
roof pitch is 7:12
?
5’
?
?º