How To Do Trigonometry Step By Step
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How To Do Trigonometry Step By Step
Trigonometry is the study of triangles, particularly right triangles. It deals with
relationships between the sides and angles of the triangles. These relationships are
expressed by the functions of sine, cosine and tangent. These functions are also
used in describing the motion of waves.
In this article, we wil be discussing the basic uses of the trigonometric functions sin, cos
and tan.
An introduction of right triangles is found in the article Pythagorean Theorem. Go check it
out if you need a bit of a refresher.
To begin with, let us define what sin, cos and tan mean. These three functions are simply
ratios of the sides of triangles that help us relate to an angle in the triangle. We'l be using
the angle A to compare these.
sin(A) = a / c (sin is the opposite side divided by the hypotenuse)
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cos(A) = b / c (cos is the adjacent side divided by the hypotenuse)
tan(A) = a / b (tan is the opposite side divided by the adjacent)
An easy way to remember this is SOHCAHTOA:
SOH, sin = opposite / hypotenuse
CAH, cos = adjacent / hypotenuse
TOA, tan = opposite / adjacent
Note that tan is not an entirely independent definition, it's just used to simplify our math. If
you were to divide sin by cos, you would get tan.
sin(A) / cos(A) = tan(A)
(a / c) / (b / c) = tan(A)
a / b = tan(A)
Based on Diagram 1.
In most cases, the notation for the angle is that of the Greek letter theta .
Solving the Trig Problems
If you are given 2 pieces of data in a triangle (i.e. if you're given an angle and a side
length or two side lengths) you can solve the entire triangle with the trig ratios and
Pythagorean Theorem.
The slight exception to this is if you're given two angles  this would basically be giving
you one piece of data since if you have one angle you can find out the other (the two
complementary angles add to 90 degrees).
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You can stil find the ratios of the side lengths using the angles that you are given and
calculating the trigonometric ratios, but they could be any set of numbers as long as they
make up that ratio. Usual y, you'll be given either two side lengths or an angle and a side
length though, and you'l be asked to solve the rest of the triangle.
Example 1 : In the first example, you are given two side lengths of 8 and 15,
respectively. We are asked to solve for the angle .What function are we supposed to
use to solve for ? With respect to , we are given the opposite and the adjacent sides.
The hypotenuse is unknown. The easiest way to tackle this is to use tan (opposite /
adjacent). We could also use the Pythagorean Theorem to solve for the hypotenuse and
then use sin or cos to solve , but why make it harder on yourself?
Now in this we don't use the tan function, we use the inverse tan function (on calculators,
it is denoted by tan^1). Since tan is used to calculate the ratio of opposite / adjacent
using , inverse tan is used to calculate using the ratio of opposite / adjacent.
It's as simple as that. Make sure your calculator is not in radians  we wil talk about that in
a future article.
Example 2 : In this example, we are given a side length of 12 and , which is equal to
30 degrees. We are asked to find the sides a and b. We know that b is the hypotenuse, a
is adjacent to , and the side length of 12 is opposite of the angle .
That's a good indication that we can use the sin function : sin (30) = 12 / b
Rearrange the equation : b = 12 / sin (30)
b = 24
Now that we have b, we can solve for a using Pythagoras, or we can use the angle
again, this time using the tan ratio.
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