Trigonometry, measurement of triangles, is used in various occupations including engineering, architecture and navigation. We start with the study of trigonometry by examining six functions. These trigonometric functions are determined by inputs that are he measures of acute angles of a right triangle. The outputs are the ratios of the length of the sides of right triangles.
If you draw a right triangle, that is a triangle with one right (90 degree) angle and two acute angles, note a base angle as Ө. We note the sides of the triangle based off this angle and the right angle. The side opposite the right angle, which is the longest side of a right triangle is the hypotenuse. The other sides are opposite of Ө and adjacent to Ө. It’s important to know the distinction between the sides because the six trigonometric functions are based on the ratio between these sides.
The trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. If you have a scientific calculator, you may have seen buttons with sin, cos and tan on them. Those are the trigonometric functions sine, cosine and tangent. Sine is the length of the side opposite Ө divided by the length of the hypotenuse. Cosine is the length of the side adjacent to Ө divided by the length of the hypotenuse. Tangent is the length of the side opposite of Ө divided by the length of the side adjacent to Ө. To figure out cotangent, secant and cosecant, they are simply the reciprocals of tangent, cosine and sine, respectively.
An important thing to note is that the value of the trigonometric functions depend only of the size of the angle Ө and not on the length of the sides. For example, a triangle with opposite side 2 and adjacent side 1 has the same trigonometric functions as a triangle with opposite side 30 and adjacent side 15, since the ratio between the two is 2:1 in both cases.
Now that we know the basic trigonometric functions, let’s evaluate them in an example. Suppose we have a right triangle with the length of the opposite side of angle Ө to be 4 and the length of the adjacent side to angle Ө to be 3. We can find the values of the six trigonometric functions once we know the length of the hypotenuse. We can get that by using the Pythagorean Theorem, which is “a squared” plus “b squared” equals “c squared”, if a, b and c are the sides of the triangle, c being the hypotenuse. Using the equation, we get c = 5. Therefore sine is opposite divided by hypotenuse, which is 4/5. Cosine is adjacent divided by hypotenuse, which is 3/5 and tangent is opposite over adjacent, which is 4/3. Cotangent is the reciprocal of tangent, so cotangent is 3/4. Secant is reciprocal of cosine, so secant is 5/3. Cosecant is reciprocal of sine, so cosecant is 5/4.
We can also find the angle measures of a right triangle using trigonometric functions. If you’ve ever noticed sin-1 , cos-1 , and tan-1 on your calculator those are the inverse sine, inverse cosine and inverse tangent buttons and are used to find the angle given a the trigonometric ratio. For example, in our above problem we know sine = 4/5, cosine = 3/5 and tangent = 4/3. Knowing these ratios we can get the value of Ө and therefore the other angles of the triangle. Take inverse sine of 4/5 and we get 53.1 degrees, which will be the same as taking inverse cosine of 3/5 and inverse tangent of 4/3. All will give the value of Ө at 53.1 degrees. Since we have a right triangle, we know one angle is 90 degrees and since the angles of a triangle add to 180 degrees, the third angle is 36.9 degrees.
These are just some of the basics with right triangle trigonometry. There is much more to learn, but understanding these basics are key to progressing to more difficult topics and applications of trigonometry.