Download Trigonometric functions The right triangle in the following diagram

Trigonometric functions
The right triangle in the following diagram has a right angle at C.
The angle of interest to us right now is angle Θ (theta). The side
opposite angle Θ is called the opposite side. The side adjacent
to angle Θ is called the adjacent side and the hypotenuse is the
side opposite the right angle (it is also the longest side in a right
triangle).
Trigonometry is based on the principle that given a right
triangle, and a fixed angle, Θ, the three sides of that triangle will
always be in the same ratio to each other. This means, for
example, no matter how big the triangle is, as long as it has a
right angle and the same angle Θ, the ratio of the
opposite/adjacent sides will always be the same value.
This is derived from the geometry of similar triangles. Two
triangles are similar if they have the same three angles, no
matter how long their sides. In the diagram below there are two
triangles that have a right angle and each has an angle of Θ = 45
degrees. One triangle is twice as big as the other in length of
sides, but they are both the same shape because their
corresponding angles are the same size. If we divide the side
opposite the 45 degree angle by the length of the side adjacent
to that angle, we get 1. And this will be true no matter how big
the triangles are. As long as the angle Θ is 45 degrees the ratio
or opposite/adjacent will be 1.
This ratio of (opposite side) / (adjacent side) has a name. It is
called the Tangent of the angle Θ and in this example it is
written as:
tan ( 45 °) = 1.
How do you say it? "The tangent of forty five degrees equals
one."
The angle Θ in a triangle can be any angle from 0° to 90°
degrees. There is an exact tangent value for every angle Θ and
you can find that value by using your calculator. We'll be doing
that in class.
There are two other important ratios in the right triangle. First
there is the Sine function. Given the same angle theta, the sine
function is the ratio of the (opposite) / (hypotenuse) and is
written
sin( Θ ) = (opposite) / (hypotenuse)
Then there is the cosine function. This is the (adjacent) /
(hypotenuse), written
cos( Θ ) = (adjacent ) / (hypotenuse )
We will cover these in class and also three other trig functions
secant, cosecant, and cotangent.
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