# Trigonometric Ratios

## Understanding Trigonometry: A Beginner's Guide to Pythagoras and Trigonometric Ratios

Have you ever looked at your calculator and wondered what those strange buttons like 'sin', 'cos', and 'tan' actually do? These buttons are crucial for understanding trigonometry, and in this article, we will delve into their meanings and how they aid us in solving mathematical problems. Before we get started, let's take a quick refresher on Pythagoras' Theorem to help solidify our understanding.

## Recap: Pythagoras vs. Trigonometry

You're probably familiar with Pythagoras' theorem already, but let's quickly recap for those who need a refresher. In this section, we will provide an overview of this concept and highlight the main differences between Pythagoras and trigonometry.

In Pythagoras' theorem, the longest side of a right-angled triangle is known as the hypotenuse. This side is always opposite the right angle and appears slanted.

## Understanding Pythagoras' Theorem

In mathematical terms, Pythagoras' theorem states that for any right-angled triangle with the hypotenuse labeled as c, and the other two sides as a and b (as shown in Figure 1), the equation a² + b² = c² is always true.

Note: The sides a and b can be switched, but c must always denote the hypotenuse.

For example, if we have a triangle with sides a, b, and c labeled as x, y, and z, and we need to find the length of side x, Pythagoras' theorem is our go-to. We label each side as shown in Figure 2, then substitute the values into the formula a² + b² = c² and solve for x, giving us the missing side's length.

In some cases, the values of the sides may result in a Pythagorean Triple, where all three sides of a right-angled triangle are whole numbers. This makes it easier to find the missing side's length.

## Exploring Trigonometry

Unlike Pythagoras' theorem, which only deals with finding the missing side of a right-angled triangle, trigonometry also helps us find missing angles. Its purpose is to solve for unknown lengths or angles in a triangle. For now, let's focus on right-angled triangles. In a future article, we will explore how trigonometry can be applied to non-right-angled triangles using the sine and cosine rules.

## Understanding Trigonometric Ratios

To grasp trigonometry, we must first define some key terms. In Figure 3, we have a right-angled triangle with one angle labeled as theta (θ). The longest side is the hypotenuse, opposite the right angle. We also have two new labels for the other sides- opposite (O) and adjacent (A).

The opposite side is the side opposite the angle, while the adjacent side is the side next to the angle that is not the hypotenuse.

## Defining Trigonometric Ratios

Now, with our triangle set up, we can define the trigonometric ratios. This is where the buttons sin, cos, and tan come in handy!

The sine (sin) of an angle is the ratio of the opposite side (O) to the hypotenuse (H).

The cosine (cos) of an angle is the ratio of the adjacent side (A) to the hypotenuse (H).

The tangent (tan) of an angle is the ratio of the opposite side (O) to the adjacent side (A).

By understanding these ratios, we can solve for missing angles or side lengths in right-angled triangles using trigonometry. With further exploration, we can apply these concepts to non-right-angled triangles as well.

## Trigonometry Examples

Now that we have covered all the necessary tools to solve trigonometric problems, let's go over some examples to better understand these concepts.

## How to Find Missing Lengths and Angles in Right-Angled Triangles using Trigonometry

In this section, we will learn how to use trigonometry to identify the missing lengths or angles in a right-angled triangle.

To begin, let's follow these steps:

**Step 1:**Label the sides of the triangle as O, A, and H.**Step 2:**Determine which sides are involved, meaning which sides do we already know and which ones we want to find.**Step 3:**Identify the appropriate trigonometric ratio.**Step 4:**Use the corresponding equation.**Step 5:**Solve for the missing side or angle.

Let's look at an example. Suppose we have a triangle where one side is labeled as x. We can find the value of x using trigonometry.

**Solution:** First, we label the sides as O, A, and H. Since we know the length of the hypotenuse and want to find the adjacent side, we use the adjacent and hypotenuse sides. Next, we identify the trigonometric ratio that involves these sides, which is cosine. Substituting into the equation, we get cos(x) = A/H. We can solve for x by multiplying both sides by H and divide by A, which gives us x = cos^{-1}A/H. Using a calculator, we get the final answer for x. Therefore, we have x = cos^{-1}A/H.

Next, let's look at finding missing angles in a right-angled triangle.

**Examples Involving Missing Angles:**

Solution: To find a missing angle, we follow similar steps as finding a missing length. Let's label the sides as O, A, and H. Then, we determine which sides are involved, which in this case is the adjacent and hypotenuse sides. Since cosine involves these sides, we use the cosine ratio to get cos(x) = A/H. To solve for x, we take the inverse cosine of both sides and get x = cos^{-1}A/H. By inputting A and H values into the calculator, we get the final answer for x. Therefore, the angle labeled x is approximately cos^{-1}A/H degrees.

## Trigonometric Ratios: Solving Right-Angled Triangles

When solving for the angle theta (θ), we must use the inverse trigonometric functions to isolate θ in the equation. The final answer should be rounded to 3 significant figures.

For example, suppose we have a triangle where one angle is labeled x. To find the value of x, we can use the appropriate trigonometric ratio by labeling the sides of the triangle. Since we know the length of the opposite side and the hypotenuse, we can use the sine function. Therefore, sin(x) = O/H = 0.5/0.8 = 0.625. To find x, we take the inverse sine of both sides, giving us x = sin^{-1}0.625. By using a calculator, we find that x is approximately 38.7 degrees.

**Trigonometric Ratios Table:**

Trigonometry is often included in calculator exams, but some values of sine, cosine, and tangent may be expected to be memorized for non-calculator exams. These values are provided in Table 1.

Now, let's consider another triangle where we need to find the length of the side labeled x, given the opposite and adjacent sides.

(Non-calculator question)

**Solution:** since we know the value of the adjacent side and want to find the opposite side, we can use the tangent function. Therefore, tan(x) = O/A = 5/4 = 1.25. We solve for x by taking the inverse tangent of both sides, giving us x = tan^{-1}1.25. Referring to Table 1, we see that tan^{-1}1.25 is approximately 51.3 degrees.

**Key Takeaways:**

- Trigonometric ratios are used to find missing sides and angles in right-angled triangles.
- The acronym SOHCAHTOA can help remember the three main trigonometric ratios: sine, cosine, and tangent.
- Label the sides of a right-angled triangle as opposite, adjacent, and hypotenuse, using the Greek letter theta (θ) to represent the angles.
- Inverse trigonometric functions can be used to find angles in a triangle.

## What Are Trigonometric Ratios?

Trigonometric ratios are ratios used to find missing sides or angles in a right-angled triangle.

## The 3 Main Trigonometric Ratios

The three main trigonometric ratios are sine, cosine, and tangent, which can be recalled using the acronym SOHCAHTOA where Sin(x) = Opposite/Hypotenuse, Cos(x) = Adjacent/Hypotenuse, and tan(x) = Opposite/Adjacent.

## Which Trigonometric Ratio is Calculated By Opposite Divided By Adjacent?

The trigonometric ratio calculated by dividing the opposite side by the adjacent side is tangent (tan).

## The Origins of Trigonometric Ratios and Their Use in Mathematics

Hipparchus, a Greek mathematician and astronomer, is widely recognized as the "father of trigonometry" and is credited with inventing the first trigonometric ratios.

## Finding Trigonometric Ratios for Any Angle

Calculating trigonometric ratios for a specific angle requires following specific guidelines. To determine the sine of an angle, the opposite side of the triangle must be divided by the hypotenuse. Similarly, the cosine can be found by dividing the adjacent side by the hypotenuse. For the tangent ratio, dividing the opposite side by the adjacent side is necessary.