How to find angles in right triangles

Every triangle has three sides, and three angles on the inside. These angles add up to 180° for every triangle, independent of the type of triangle. In a right triangle, one of the angles is exactly 90°. Such an angle is called a right angle.

To calculate the other angles we need the sine, cosine and tangent. In fact, the sine, cosine and tangent of an acute angle can be defined by the ratio between sides in a right triangle.

Right Triangle

Just like every other triangle, a right triangle has three sides. One of them is the hypothenuse, which is the side opposite to the right angle. The other two sides are identified using one of the other two angles. The other angles are formed by the hypothenuse and one other side. This other side is called the adjacent side. Then, there is one side left which is called the opposite side. When you would look from the perspective of the other angle the adjacent and opposite side are flipped.

So if you look at the picture above, then the hypothenuse is denoted with h. When we look from the perspective of the angle alpha the adjacent side is called b, and the opposite side is called a. If we would look from the other non-right angle, then b is the opposite side and a would be the adjacent side.

Sine, Cosine and Tangent

The sine, cosine and tangent can be defined using these notions of hypothenuse, adjacent side and opposite side. This only defines the sine, cosine and tangent of an acute angle. The sine, cosine and tangent are also defined for non-acute angles. To give the full definition, you will need the unit circle. However, in a right triangle, all angles are non-acute, and we will not need this definition.

The sine of an acute angle is defined as the length of the opposite side divided by the length of the hypothenuse.

The cosine of an acute angle is defined as the length of the adjacent side divided by the length of the hypothenuse.

The tangent of an acute angle is defined as the length of the opposite side divided by the length of the adjacent side.

Or more clearly formulated:

• sin(x) = opposite/hypothenuse
• cos(x) = adjacent/hypothenuse
• tan(x) = opposite/adjacent

Calculating an Angle in a Right Triangle

The rules above allow us to do calculations with the angles, but to calculate them directly we need the inverse function. An inverse function f-1 of a function f has as input and output the opposite of the function f itself. So if f(x) = y then f-1(y) = x.

So if we know sin(x) = y then x = sin-1(y), cos(x) = y then x = cos-1(y) and tan(x) = y then tan-1(y) = x. Since these functions come up a lot they have special names. The inverse of the sine, cosine and tangent are the arcsine, arccosine and arctangent.

For more information on inverse functions and how to calculate them, I recommend my article about the inverse function.

• Math: How to Find the Inverse of a Function
  

A right triangle with sides r, y and x

An Example of Calculating the Angles in a Triangle

In the triangle above we are going to calculate the angle theta. Let x = 3, y = 4. Then by the Pythagorean theorem we know that r = 5, since sqrt(32 + 42) = 5. Now we can calculate the angle theta in three different ways.

sin(theta) = y/r = 3/5

cos(theta) = x/r = 4/5

tan(theta) = y/x = 3/4

So theta = arcsin(3/5) = arccos(4/5) = arctan(3/4) = 36.87°. This allows us to calculate the other non-right angle as well, because this must be 180-90-36.87 = 53.13°. This is because the sum of all angles of a triangle always is 180°.

We can check this using the sine, cosine and tangent again. We call the angle alpha then:

sin(alpha) = x/r = 4/5

cos(alpha) = y/r =3/5

tan(alpha) = y/x = 4/3

Then alpha = arcsin(4/5) = arccos(3/5) = arctan(4/3) = 53.13. So this is indeed equal to the angle we calculated with the help of the other two angles.

We can also do it the other way around. When we know the angle and the length of one side, we can calculate the other sides. Let's say we have a slide which is 4 meters long and goes down at an angle of 36°. Now we can calculate how much vertical and horizontal space this slide will take. We are basically in the same triangle again, but now we know theta is 36° and r = 4. Then to find the horizontal length x we can use the cosine. We get:

cos(36) = x/4

And therefore x = 4*cos(36) = 3.24 meters.

To calculate the height of the slide we can use the sine:

sin(36) = y/4

And therefore y = 4*sin(36) = 2.35 meters.

Now we can check whether tan(36) is indeed equal to 2.35/3.24. We find tan(36) = 0.73, and also 2.35/3.24 = 0.73. So indeed we did everything correctly.

A right triangle in everyday life

The Secant, Cosecant and Cotangent

The sine, cosine and tangent define three ratios between sides. There are however three more ratios we could calculate. If we divide the length of the hypothenuse by the length of the opposite is the cosecant. Dividing the hypothenuse by the adjacent side gives the secant and the adjacent side divided by the opposite side results in the cotangent.

This means that these quantities can be directly calculated from the sine, cosine and tangent. Namely:

sec(x) = 1/cos(x)

cosec(x) = 1/sin(x)

cot(x) = 1/tan(x)

The secant, cosecant and cotangent are used very rarely used, because with the same inputs we could also just use the sine, cosine and tangent. Therefore, a lot of people would not even know they exist.

The Pythagorean Theorem

The Pythagorean Theorem is closely related to the sides of right triangles. It is very well known as a2 + b2 = c2. I wrote an article about the Pythagorean Theorem in which I went deep into this theorem and its proof.

• Math: The Pythagorean Theorem
  

What You Need to Determine Everything in a Triangle

We can calculate the angle between two sides of a right triangle using the length of the sides and the sine, cosine or tangent. To do this, we need the inverse functions arcsine, arccosine and arctangent. If you only know the length of two sides, or one angle and one side, this is enough to determine everything in the triangle.

Instead of the sine, cosine and tangent, we could also use the secant, cosecant and cotangent, but in practice, these are hardly ever used.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.