The First Steps in Learning Trigonometry - Trigonometric Functions
Diagram 1 - Right Angle Triangle
What is Trigonometry?
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 will 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'll be using the angle A to compare these.
sin(A) = a / c (sin is the opposite side divided by the hypotenuse)
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 Θ.
Solving the Trig Problems
The main trick with trigonometry 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). You can still 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.
Usually, you'll be given either two side lengths or an angle and a side length though, and you'll be asked to solve the rest of the triangle.
In the first example, you are given two side lengths of 8 and 15, respectively. We are asked to solve for the angle Θ.What function are we supposed to use to solve for Θ?
With respect to Θ, 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 Θ, 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 tanΘ is used to calculate the ratio of opposite / adjacent using Θ, inverse tan is used to calculate Θ using the ratio of opposite / adjacent.
inverse tan (8/15) = 28.07 degrees
It's as simple as that. Make sure your calculator is not in radians - we will talk about that in a future article.
Notice that inverse cos (15/17) = inverse sin (8/17) = inverse tan (8/15) = 28.07 degrees.
In this example, we are given a side length of 12 and Θ, 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 Θ, and the side length of 12 is opposite of the angle Θ.
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 Θ again, this time using the tan ratio.
tan (30) = 12 / a
a = 12 / tan (30)
a = 20.8
You now have the basics to solve trigonometric ratios consisting of sin, cos and tan.
Now go get some practice!