Download Sin (x) = opposite/hypotenuse

Trigonometric Functions:
Trigonometric functions are ratios that describe the relationship between sides of a
right triangle (and a right triangle ONLY) with respect to a given angle.
Hypotenuse
Opposite
Adjacent
In the picture above, the angle denotated with the curve is the angle we are observing.
In this picture, the there are three sides described in relation to this angle. There is
the one opposite from it,
A side that is adjacent to it,
And the side that is the hypotenuse of the triangle.
Trigonometric functions are ratios of these sides to describe the angle through which
they are related.
Sine
The trig function sine of x (notated “sin (x)”) reefers to the ratio of the Opposite side
over the Hypotenuse, when “x” is the angle being described.
In other words, sin (x) = (opposite/hypotenuse)
Example 1:
6
10
8
The sin (x) = 6/10 or 3/5 or .6
Example 2:
10
1
3
The sin (x) = 3/10
Cosine
The trig function cosine of x (notated “cos (x)”) reefers to the ratio of the Adjacent
side over the Hypotenuse, when “x” is the angle being described.
Example 1:
6
10
8
The cos (x) = 8/10 or 4/5 or .8
Example 2:
10
1
3
The cos (x) = 1/10
Tangent
The trig function tangent of x (notated “tan (x)”) reefers to the ratio of the Opposite
side over the Adjacent side, when “x” is the angle being described.
Example 1:
6
10
8
The tan (x) = 6/8 or ¾ or .75
Example 2:
10
1
The tan (x) = 3/1 or just 3.
3
So why is this important? Say we’re given a triangle that only has the right angle marked
(like all of the ones above)… how can we figure out what the other angles are? Well, we
know they have to be complimentary, right? But what else do we know about them?
Because of trig functions, if we are given just the lengths of the sides of the triangle, we
can figure out what each angle is.
Just like any other mathematical process, trigonometric functions have inverses (just like
addition’s inverse is subtraction and multiplication’s inverse is division). These inverses
are notated respectively as arcsine, arccosine and arctangent. These operations “undo” the
relationship between the sides that describe the angle and turn it into the angle that is the
relationship between the sides.
We are going to do this using TI-83 Calculators.
Example 1:
b
6
8
a
What are the sine, cosine and tangent of angle a?
First, we need to use the Pythagorean Theorem to find the hypotenuse.
(6^2) + (8^2) = 100
(c^2) = 100
c = 10
sin (a) = 6/10
cos (a) = 8/10
tan (a) = 6/8
If we do the inverse of each one of these, what do we get?
- Press the blue “2nd” button then hit the “sin” button, punch in “6/10”
then hit “Enter”.
- Press the blue “2nd” button, hit the “cos” button, punch in 8/10 then hit
“Enter”
- Press the blue “2nd” button, hit the “tan” button, punch in 6/8 then hit “Enter”
You should have come up with these values:
-- a = ((sin ^ -1)(.6))
a = 36.87
-- a = ((cos^-1)(.8))
a = 36.87
-- a = ((tan^-1)(.75)
a = 36.87
This shows us that this angle “a” is 36.87
*Note: you don’t need to do out all three inverse trigonometric function. Just
doing one will give you the angle.
What are the sine, cosine and tangent of angle b?
sin (b) = 8/10
cos (b) = 6/10
tan (b) = 8/6
Do the respective inverse of each funciton:
-- b = ((sin^-1)(8/10))
b = 53.13
-- b = ((cos^-1)(6/10))
b = 53.13
-- b = ((tan^-1)(8/6))
b = 53.13
This tells us that angle “b” measures 53.13.
How else could we have figured this out? We know that angles a and b must be
complimentary, so we could have found angle b by subtracting 36.87 from 180 which
would have given us the same answer that angle b = 53.13.
Example 2 (try it on your own):
a
10
1
3
Find angles a and b.
tan (a) = 3/1
a = ((tan^-1)(3)
a = 71.56
tan (b) = 1/3
b = ((tan^-1)(1/3))
b = 18.44
b