Explainer: Trigonometric Ratios in Right Triangles

In this explainer, we will learn how to find and express the values of the three trigonometric ratios—sine, cosine, and tangent—for a given angle in a right triangle.

A useful skill when working with right triangles is having the ability to calculate the different trigonometric ratios for a specified angle. Recall the acronym “SOH CAH TOA.” This can be very useful in remembering the three different trigonometric ratios.

Trigonometric Ratios

O stands for the “opposite,” A stands for the “adjacent,” H stands for the “hypotenuse,” and 𝜃 is the angle.

We can use the trigonometric ratios in combination with the Pythagorean theorem to calculate all the trigonometric ratios for a specified angle. We will demonstrate how to do this using a series of examples.

Before we start, let us recap how to correctly label a right triangle relative to a specified angle. The opposite side is the side directly opposite the angle, the hypotenuse is the longest side (which is always opposite the right angle), and the adjacent is the side next to the angle (which is not the hypotenuse). We can demonstrate this using a diagram.

It is helpful to recall the Pythagorean theorem. In words, it states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. For a triangle where the hypotenuse has length 𝑐 and the legs (the other two sides) have lengths 𝑎 and 𝑏, the Pythagorean theorem can be represented algebraically as 𝑎+𝑏=𝑐.

Now, let us look at the first example.

Example 1: Finding the Values of the Three Trigonometric Ratios of an Angle

Find the main trigonometric ratios of 𝐵 given that 𝐴𝐵𝐶 is a right triangle at 𝐶, where 𝐴𝐵=30cm and 𝐵𝐶=18cm.

Answer

Our first step in answering this question should be to draw a diagram of the triangle.

We then need to calculate the length of 𝐴𝐶. We can do this using the Pythagorean theorem. We label the hypotenuse 𝑐 and the legs 𝑎 and 𝑏. So, we have that 𝐴𝐵=𝑐, 𝐴𝐶=𝑏, and 𝐵𝐶=𝑎. Note here that the order of 𝑎 and 𝑏 does not matter as long as we are consistent throughout our calculation. Recall that 𝑎+𝑏=𝑐.

If we substitute the values of 𝑎, 𝑏, and 𝑐, we get 18+𝑏=30.

We then subtract 18 from each side and take square roots to find that 𝑏=3018.

Calculating this, we find that 𝑏=24, so 𝐴𝐶=24.

We now need to add this length to our diagram and label the opposite, hypotenuse, and adjacent sides, relative to 𝐵.

We can then use the acronym SOH CAH TOA to help us remember the trigonometric ratios and then we can use these to calculate the values of sin𝐵, cos𝐵, and tan𝐵.

By substituting the values of O, A, and H into the trigonometric ratios and simplifying the resulting fractions, we find that sincosandtan𝐵=45,𝐵=34,𝐵=43.

Now, let us look at a question that contains an additional context and that adds an additional layer of difficulty.

Example 2: Finding the Values of Trigonometric Ratios for an Angle given an Additional Context

𝐴𝐵 is a diameter of a circle with radius 62.5 cm. Point 𝐶 is on the circumference of the circle where 𝐴𝐶𝐶𝐵 and 𝐴𝐶=75cm. Find the exact values of cos𝐴 and sin𝐵.

Answer

The first thing to note here is that 𝐴𝐵 is a diameter and hence has a length that is twice the radius of the circle: 62.5×2=125. It is then helpful to draw the triangle using these lengths.

We then need to calculate the unknown length 𝐶𝐵 using the Pythagorean theorem. Let us call 𝐴𝐶=𝑎, 𝐵𝐶=𝑏, and the hypotenuse 𝐴𝐵=𝑐. Recall that 𝑎+𝑏=𝑐.

Substituting the lengths, we find that 75+𝑏=125.

If we then subtract 75 from each side and take square roots, we get 𝑏=12575.

Calculating this, we find that 𝑏=100, so 𝐵𝐶=100.

The question asks us to calculate cos𝐴 and sin𝐵. We need to label the triangle in terms of the opposite, adjacent, and hypotenuse, first relative to 𝐴 and then relative to 𝐵. When we label the triangle relative to 𝐴, we get the following.

We can then use the acronym SOH CAH TOA to help us remember the trigonometric ratios.

Using these and the labeled triangle, we can calculate the value of cos𝐴: cos𝐴=75125=35.

If we then label the triangle relative to 𝐵, we get the following.

Then, as before, we can calculate the value of sin𝐵: sin𝐵=75125=35.

In some questions, you will be given a specific trigonometric ratio and asked to calculate the value of others. To conclude, we will look at a couple of examples where this is the case.

Example 3: Finding the Values of Trigonometric Ratios from a Known Ratio

The value of tan𝜃=45. Find the exact values of sin𝜃 and cos𝜃.

Answer

To solve this problem, it is helpful to start by drawing a right triangle containing an angle 𝜃 with labels for the opposite, adjacent, and hypotenuse, relative to 𝜃.

We can then use the fact that tanOA𝜃= to find that the opposite side has length 4 and the adjacent side has length 5. If we then call the length of the hypotenuse 𝑥, we can find this using the Pythagorean theorem: 𝑥=4+5.

Taking square roots, we find that 𝑥=4+5.

Simplifying this, we find that 𝑥=41.

We will leave this in this form as the question asks for our answers in an exact form. We can now label the lengths of the triangle.

We can then recall the trigonometric ratios for sine and cosine, sinOHandcosAH𝜃=𝜃=, and use these, along with our triangle, to calculate our answers: sinandcos𝜃=441𝜃=541.

Example 4: Finding the Values of Trigonometric Ratios from the Tangent Ratio

Find the main trigonometric ratios of 𝐶 given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where 20𝐴=21tan.

Answer

To solve this problem, it is helpful to start by drawing a right triangle. We are told in the question that 20𝐴=21tan. We can rearrange this to give us tan𝐴=2120, and we can compare this to the general tangent ratio tanOppAdj𝐴=.

We find that the side opposite to angle 𝐴 has length 21 and the side adjacent to angle 𝐴 has length 20.

We can now find the length of the hypotenuse using the Pythagorean theorem. If we call the length of the hypotenuse 𝑥, we have 𝑥=20+21.

Taking square roots, we find that 𝑥=20+21.

Simplifying this, we find that 𝑥=29.

We can now label the hypotenuse in our diagram and label the sides relative to 𝐶.

We can then recall the trigonometric ratios for sine and cosine, sinOppHypandcosAdjHyp𝜃=𝜃=, and use these, along with our triangle and tangent ratio, to calculate our answers: sincosandtan𝐶=2029,𝐶=2129,𝐶=2021.

The methods outlined in these latter two examples become very useful when working with questions in mechanics that involve slopes or ramps. The angle of the slope is often given as a tangent ratio of its length and height, and calculations will often require the use of the sine and cosine ratios.

Let us look at one more example where we need to use the trigonometric ratios to help solve the problem.

Example 5: Using Trigonometric Ratios to Solve Problems

Find 𝑚𝐴 given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where 2𝐴𝐵=𝐴𝐶.

Answer

From the question, we are told that the triangle 𝐴𝐵𝐶 is a right triangle at 𝐵 and we are asked to calculate the measure of angle 𝐴. We can use this information to draw a diagram, labeling the triangle with respect to angle 𝐴:

We are also told information about lengths 𝐴𝐵 and 𝐴𝐶, specifically, that 2𝐴𝐵=𝐴𝐶.

Rearranging, we can see that 2=𝐴𝐶𝐴𝐵.

Looking at the diagram, we can see that this is a ratio between the hypotenuse and the adjacent sides of the triangle. Recall that the cosine ratio is cosAdjHyp𝜃=, which, for this triangle, is cos𝜃=𝐴𝐵𝐴𝐶.

This is the reciprocal of what we were given in the question. That is, 𝐴𝐵𝐴𝐶=12.

Therefore, cos𝜃=12, which means 𝜃=12=45.arccos

Key Points

To find the values of the three trigonometric ratios of an angle, remember the following steps:

  1. When working with right triangles, we use the terms opposite, adjacent, and hypotenuse to refer to the sides of the triangle. The hypotenuse is always opposite the right angle and is the longest side. The opposite and adjacent are labeled in relationship to a given angle often denoted 𝜃. The adjacent is the side next to the angle 𝜃 which is not the hypotenuse. As for the opposite, it is the last side of the triangle. It is called the opposite since it is opposite the given angle.
  2. Recall the acronym “SOH CAH TOA,” where O stands for the opposite, A stands for the adjacent, H stands for the hypotenuse, and 𝜃 is the angle. The trigonometric ratios are sinOHcosAHandtanOA𝜃=,𝜃=,𝜃=.
  3. We can use the trigonometric ratios in combination with the Pythagorean theorem, 𝑎+𝑏=𝑐, to calculate all the trigonometric ratios for a specified angle.

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