The Bare Basics of Trigonometry: A Simple Review

Trigonometry can look intimidating at first, but the truth is that the basic ideas are very manageable once you understand what the parts of a right triangle mean and which formula to use.

For many students, Grade 11 trigonometry starts with two core skills:

1. Using the Pythagorean Theorem

2. Using SOH CAH TOA

These are the foundations that students continue using throughout high school math. If these basics are clear, later topics become much easier.

What Is Trigonometry?

Trigonometry is the part of math that studies the relationship between the angles and side lengths of triangles.

At the beginner level, trigonometry usually focuses on right triangles. A right triangle is a triangle that has one 90 degree angle.

In a right triangle, the sides have special names:

• The hypotenuse is always the longest side, and it is always across from the 90 degree angle.

• The opposite side is the side across from the angle you are focusing on.

• The adjacent side is the side next to the angle you are focusing on, not counting the hypotenuse.

Understanding these names is extremely important, because SOH CAH TOA depends on them.

Part 1: The Pythagorean Theorem

The Pythagorean Theorem applies to right triangles only.

The formula is:

a² + b² = c²

where:

• a and b are the two shorter sides

• c is the hypotenuse

This formula lets you find a missing side when you already know the other two sides.

Example 1: Finding the Hypotenuse

A right triangle has side lengths of 6 cm and 8 cm. Find the hypotenuse.

Step 1: Identify what you know

The two shorter sides are:

• a = 6

• b = 8

We want to find the hypotenuse, so:

• c = ?

Step 2: Substitute into the formula

a² + b² = c²

6² + 8² = c²

Step 3: Square the known values

36 + 64 = c²

100 = c²

Step 4: Take the square root of both sides

c = √100

c = 10

Final Answer

The hypotenuse is 10 cm.

Example 2: Finding a Non-Hypotenuse Side

A right triangle has a hypotenuse of 13 m and one leg of 5 m. Find the other leg.

Step 1: Identify what you know

• c = 13

• a = 5

• b = ?

Step 2: Use the Pythagorean Theorem

a² + b² = c²

5² + b² = 13²

Step 3: Square the known values

25 + b² = 169

Step 4: Isolate b²

b² = 169 - 25

b² = 144

Step 5: Take the square root

b = √144

b = 12

Final Answer

The missing side is 12 m.

Part 2: SOH CAH TOA

SOH CAH TOA is a memory tool that helps students remember the three main trigonometric ratios.

SOH:

sin(θ) = opposite / hypotenuse

CAH:

cos(θ) = adjacent / hypotenuse

TOA:

tan(θ) = opposite / adjacent

Here, the symbol θ represents the angle you are working with.

The most important part is choosing the correct sides based on that angle.

Before Using SOH CAH TOA

Always follow these steps:

1. Identify the angle given.

2. Label the sides relative to that angle.

3. Decide whether you need sine, cosine, or tangent.

4. Substitute carefully.

5. Solve.

6. Make sure your calculator is in degree mode unless the question says otherwise.

Example 3: Using SOH CAH TOA to Find a Side

A ladder leans against a wall. The ladder makes an angle of 60 degrees with the ground. If the ladder is 10 m long, how high up the wall does it reach?

Step 1: Picture the triangle

The wall, ground, and ladder form a right triangle.

• The ladder is the hypotenuse because it is the longest side.

• The height up the wall is opposite the 60 degree angle.

• The ground is adjacent to the 60 degree angle.

We are trying to find the opposite side, and we know the hypotenuse.

Step 2: Choose the correct trig ratio

Since we need opposite and hypotenuse, we use sine.

sin(θ) = opposite / hypotenuse

Step 3: Substitute the values

sin(60) = opposite / 10

Step 4: Solve for the opposite side

Multiply both sides by 10:

opposite = 10 sin(60°)

Using a calculator:

opposite ≈ 10(0.866)

opposite ≈ 8.66

Final Answer

The ladder reaches about 8.66 m up the wall.

Example 4: Using SOH CAH TOA to Find an Angle

A right triangle has an opposite side of 7 cm and an adjacent side of 24 cm. Find the angle θ.

Step 1: Decide which ratio to use

We know:

• opposite = 7

• adjacent = 24

Since we have opposite and adjacent, we use tangent.

tan(θ) = opposite / adjacent

Step 2: Substitute the values

tan(θ) = 7 / 24

tan(θ) = 0.2917

Step 3: Use the inverse tangent function

To find the angle itself, we use tan⁻¹ on the calculator.

θ = tan⁻¹(0.2917)

θ ≈ 16.3°

Final Answer

The angle is approximately 16.3 degrees.

Common Mistakes in Basic Trigonometry

A lot of trigonometry mistakes come from a few common issues:

Mixing up the side names

The opposite and adjacent sides depend on which angle you are using. They can change if you switch to a different angle.

Forgetting that the hypotenuse is always across from the right angle

The hypotenuse never changes. It is always the longest side in a right triangle.

Using the wrong trig ratio

Always ask yourself: which two sides do I know, and which side am I trying to find?

Forgetting inverse trig for angles

If you are solving for an angle, you usually need sin⁻¹, cos⁻¹, or tan⁻¹.

Calculator not in degree mode

This is a very common source of wrong answers in high school trigonometry.

Final Review

If you are reviewing the absolute basics of trigonometry, here is what you should know:

• The Pythagorean Theorem helps you find a missing side in a right triangle.

• SOH CAH TOA helps you relate side lengths and angles in a right triangle.

• The hypotenuse is always across from the 90 degree angle.

• The opposite and adjacent sides depend on the angle you are focusing on.

• To find an angle, use an inverse trig function.

• Make sure your calculator is in degree mode.

These ideas form the foundation of basic trigonometry, and students who are comfortable with them are in a much better position to succeed in higher level trigonometry.

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