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Trigonometric Ratios

Q: Can a trigonometric ratio be greater than 1?

Yes, the Tangent ratio can be greater than $1$. This happens when the opposite side is longer than the adjacent side. However, the Sine and Cosine ratios can never be greater than $1$ because the hypotenuse is always the longest side.

Q: What does the $\sin^{-1}$ button on my calculator do?

The $\sin^{-1}$ button stands for 'inverse sine'. You use it when you know the ratio of the opposite side to the hypotenuse, and you want to find the angle that creates that ratio. It essentially works backwards to find the angle.

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Ever wondered how people measure the height of a giant tree or a tall building without a huge ruler? They use a special kind of math called trigonometry! It’s all about the secret relationships between the angles and side lengths in right triangles, and it's a powerful tool for solving real-world puzzles.

Trigonometric Ratios — an original Algebra911 reference diagram defining trigonometric ratios with its key formula and a worked example. — Trigonometric Ratios Explained

What Are Trigonometric Ratios?

Trigonometric ratios are special measurements that describe the relationship between the angles and the lengths of the sides in a right-angled triangle. A right-angled triangle (or just right triangle) is any triangle that has one perfect $90$ -degree angle, like the corner of a square. These ratios are like secret codes that connect an angle's size to the length of the triangle's sides. If you know one angle and one side length, these ratios can help you figure out the lengths of the other sides without ever measuring them!

Think of them as a powerful toolkit for solving puzzles involving heights, distances, and angles. For example, you can figure out the height of a flagpole just by measuring its shadow and the angle of the sun. The three most important and most common trigonometric ratios are called Sine, Cosine, and Tangent. Each one is a specific fraction, or ratio, comparing the lengths of two of the three sides of a right triangle. By understanding these three ratios, you unlock the ability to solve a whole new world of math problems.

What Are the Special Parts of a Right Triangle?

Before we can use our trigonometric ratios, we need to know the names for the three sides of a right triangle. Their names change depending on which angle we are focusing on. Let's pick one of the non-right angles and call it our 'reference angle', which we'll label with the Greek letter theta, $θ$ .

Here are the three sides:

Hypotenuse: This one is the easiest to find. It is ALWAYS the longest side of the right triangle, and it is ALWAYS the side directly across from the $90$ -degree right angle. It doesn't matter which reference angle you choose; the hypotenuse is always the same.
Opposite Side: This is the side that is directly across from your reference angle ( $θ$ ). It's the one side that doesn't touch the angle at all. Imagine you are standing at the corner of the angle $θ$ ; the opposite side is the wall on the other side of the room.
Adjacent Side: The word 'adjacent' means 'next to'. This is the side that is next to your reference angle ( $θ$ ), but it is NOT the hypotenuse. It's one of the two sides that forms the angle itself.

Remember, the names 'Opposite' and 'Adjacent' depend completely on which angle you choose as your reference! If you switch to the other non-right angle, the opposite and adjacent sides will swap places.

How Do You Remember the Ratios with SOH CAH TOA?

Trying to remember which sides go with which ratio can be tricky. Luckily, there's a very famous and easy-to-remember mnemonic: SOH CAH TOA. It sounds like a silly phrase, but it's the key to mastering trigonometry. Each part of the phrase tells you exactly how to set up one of the ratios.

Let's break it down:

SOH: Sine is Opposite over Hypotenuse

This tells us that the Sine of an angle (written as $\sin (θ)$ ) is the ratio of the length of the Opposite side to the length of the Hypotenuse.

\sin (θ) = \frac{Opposite}{Hypotenuse}

CAH: Cosine is Adjacent over Hypotenuse

This tells us that the Cosine of an angle (written as $\cos (θ)$ ) is the ratio of the length of the Adjacent side to the length of the Hypotenuse.

\cos (θ) = \frac{Adjacent}{Hypotenuse}

TOA: Tangent is Opposite over Adjacent

Finally, this tells us that the Tangent of an angle (written as $\tan (θ)$ ) is the ratio of the length of the Opposite side to the length of the Adjacent side.

\tan (θ) = \frac{Opposite}{Adjacent}

By simply remembering 'SOH CAH TOA', you hold the key to all three basic trigonometric ratios!

How Do You Find a Missing Side Length?

This is where trigonometry becomes a superpower! If you know the measure of one angle (besides the right angle) and the length of one side, you can find the length of any other side. Here's how:

Identify your knowns and unknowns: What angle do you know? What side length do you know? What side length are you trying to find?
Label the sides: From the perspective of your known angle, label the sides of the triangle as Opposite, Adjacent, and Hypotenuse.
Choose the right ratio: Look at the side you know and the side you want to find. Which ratio from SOH CAH TOA uses those two sides? If you have the Opposite and need the Hypotenuse, you'll use Sine (SOH). If you have the Adjacent and need the Opposite, you'll use Tangent (TOA).
Set up the equation and solve: Write down the formula for your chosen ratio, plug in the numbers you know, and solve for the unknown side. You will need a scientific calculator to find the value of $\sin$ , $\cos$ , or $\tan$ for the angle.

Example 1

You are flying a kite. The kite string is $150$ meters long and it makes a $40$ -degree angle with the ground. How high is the kite in the air? Let's call the height $h$ .

1. Knowns/Unknowns: We know the angle ( $40^{\circ}$ ) and the length of the string (the hypotenuse = $150$ m). We want to find the height ( $h$ ), which is the side opposite the angle.

2. Label Sides: The string is the Hypotenuse. The height $h$ is Opposite the $40^{\circ}$ angle.

3. Choose Ratio: We have Opposite and Hypotenuse. Looking at SOH CAH TOA, we see that SOH uses Opposite and Hypotenuse. So, we'll use Sine.

4. Set up and Solve:
The formula is $\sin (θ) = \frac{Opposite}{Hypotenuse}$ .
Plug in our values: $\sin (40^{\circ}) = \frac{h}{150}$ .
To solve for $h$ , we multiply both sides by $150$ : $h = 150 \times \sin (40^{\circ})$ .
Now, use a calculator to find $\sin (40^{\circ})$ , which is approximately $0.6428$ .
$h = 150 \times 0.6428 \approx 96.42$ .
Answer: The kite is approximately $96.42$ meters high.

Example 2

You are standing $50$ feet away from the base of a tall building. You look up to the top of the building at an angle of $65$ degrees. How tall is the building? Let's call the height $x$ .

1. Knowns/Unknowns: We know the angle ( $65^{\circ}$ ) and the distance from the building (the adjacent side = $50$ ft). We want to find the building's height ( $x$ ), which is the side opposite the angle.

2. Label Sides: The height $x$ is Opposite the $65^{\circ}$ angle. The distance on the ground is Adjacent to the angle.

3. Choose Ratio: We have Opposite and Adjacent. Looking at SOH CAH TOA, we see that TOA uses Opposite and Adjacent. We'll use Tangent.

4. Set up and Solve:
The formula is $\tan (θ) = \frac{Opposite}{Adjacent}$ .
Plug in our values: $\tan (65^{\circ}) = \frac{x}{50}$ .
To solve for $x$ , we multiply both sides by $50$ : $x = 50 \times \tan (65^{\circ})$ .
Use a calculator to find $\tan (65^{\circ})$ , which is approximately $2.1445$ .
$x = 50 \times 2.1445 \approx 107.23$ .
Answer: The building is approximately $107.23$ feet tall.

How Can You Find a Missing Angle?

What if you know the side lengths but need to find an angle? For this, we use the 'inverse' trigonometric functions. They are like the 'undo' button for Sine, Cosine, and Tangent. On your calculator, they look like $\sin^{- 1}$ , $\cos^{- 1}$ , and $\tan^{- 1}$ . They take a ratio of sides as input and give you back the angle.

The steps are very similar to finding a missing side:

Identify your knowns and unknowns: Which two side lengths do you know? Which angle are you trying to find?
Label the sides: From the perspective of your unknown angle ( $θ$ ), label the two sides you know (e.g., Opposite and Adjacent, Adjacent and Hypotenuse, etc.).
Choose the right ratio: Look at the two sides you know. Which ratio from SOH CAH TOA uses them?
Set up the ratio and use the inverse function: Write the ratio, then use the corresponding inverse function to solve for the angle $θ$ .

Example 3

A wheelchair ramp is $20$ feet long and rises to a platform that is $2$ feet off the ground. What is the angle the ramp makes with the ground? Let's call the angle $A$ .

1. Knowns/Unknowns: We know the length of the ramp (the hypotenuse = $20$ ft) and the height it rises (the opposite side = $2$ ft). We want to find the angle $A$ .

2. Label Sides: The ramp's length is the Hypotenuse. The height is Opposite the angle $A$ .

3. Choose Ratio: We have Opposite and Hypotenuse, so we use Sine (SOH).

4. Set up and Solve:
The formula is $\sin (A) = \frac{Opposite}{Hypotenuse}$ .
Plug in our values: $\sin (A) = \frac{2}{20}$ .
Simplify the fraction: $\sin (A) = 0.1$ .
Now, to find the angle $A$ , we use the inverse sine function: $A = \sin^{- 1} (0.1)$ .
Use a calculator: $A \approx {5.74}^{\circ}$ .
Answer: The ramp makes an angle of approximately $5.74$ degrees with the ground.

Key formulas for trigonometric ratios by Algebra911.

What Are Some Common Mistakes to Avoid?

Trigonometry is very powerful, but it's easy to make small mistakes. Here are a few common pitfalls to watch out for:

Mixing up Opposite and Adjacent: This is the most common error. Always remember to label your sides from the perspective of the reference angle you are using in the problem. What is 'opposite' for one angle is 'adjacent' for the other!
Using the Wrong Ratio: Double-check SOH CAH TOA to make sure you've selected the right tool for the job. If you have the Adjacent side and Hypotenuse, you must use Cosine (CAH), not Sine or Tangent.
Calculator in the Wrong Mode: Scientific calculators can measure angles in Degrees or Radians. For the problems you'll be doing, you must make sure your calculator is in Degree (DEG) mode. If it's in Radian (RAD) mode, you will get the wrong answers. Look for a 'DRG' or 'MODE' button to change this setting.
Applying Ratios to Non-Right Triangles: SOH CAH TOA and the basic trigonometric ratios only work for right-angled triangles. If your triangle doesn't have a $90$ -degree angle, you cannot use these simple formulas.

Quick Summary and Reference Table

Trigonometry might seem complex, but it all comes down to a few key ideas. It's the study of right triangles, connecting angles to side lengths. The most important thing to remember is the mnemonic SOH CAH TOA, which tells you how to set up the three main ratios: Sine, Cosine, and Tangent. With these tools and a calculator, you can find missing sides or angles in any right triangle.

Here is a handy reference table to help you remember the ratios:

Ratio Name	Abbreviation	Formula	Mnemonic
Sine	$\sin (θ)$	$\frac{Opposite}{Hypotenuse}$	SOH
Cosine	$\cos (θ)$	$\frac{Adjacent}{Hypotenuse}$	CAH
Tangent	$\tan (θ)$	$\frac{Opposite}{Adjacent}$	TOA

Frequently Asked Questions

Do trigonometric ratios work for all triangles?

No, the basic trigonometric ratios of Sine, Cosine, and Tangent (SOH CAH TOA) only work for right-angled triangles. This is because their definitions are based on the relationships between the sides relative to a $90$ -degree angle.

What does SOH CAH TOA stand for?

SOH CAH TOA is a mnemonic to help you remember the three basic trig ratios. It stands for: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.

Do I need a special calculator for trigonometry?

Yes, you will need a scientific calculator. These calculators have the special buttons for Sine (sin), Cosine (cos), and Tangent (tan), as well as their inverse functions ( $\sin^{- 1}$ , $\cos^{- 1}$ , $\tan^{- 1}$ ) that you need to solve these problems.

What is the difference between the opposite and adjacent sides?

The labels 'opposite' and 'adjacent' depend entirely on which angle you are using as your reference. The opposite side is always across from the angle, while the adjacent side is always next to the angle (but is not the hypotenuse).

Why is the hypotenuse always the longest side?

In any triangle, the longest side is always across from the largest angle. In a right triangle, the largest angle is the right angle ( $90$ degrees), so the side across from it, the hypotenuse, must be the longest side.

Can a trigonometric ratio be greater than 1?

Yes, the Tangent ratio can be greater than $1$ . This happens when the opposite side is longer than the adjacent side. However, the Sine and Cosine ratios can never be greater than $1$ because the hypotenuse is always the longest side.

What does the

\sin^{- 1}

button on my calculator do?

The $\sin^{- 1}$ button stands for 'inverse sine'. You use it when you know the ratio of the opposite side to the hypotenuse, and you want to find the angle that creates that ratio. It essentially works backwards to find the angle.