Trigonometry – JSS3 Mathematics Lesson Note (Second Term)

Lesson objectives

By the end of this lesson, students will be able to:

• Understand what trigonometry means
• Know the parts of a right triangle
• Learn the three main trigonometry rules (sine, cosine, tangent)
• Use simple trigonometry to solve problems
• Find trigonometry in everyday life

What is trigonometry?

Trigonometry is the study of triangles and angles. The word comes from Greek words that mean “triangle measuring.” It helps us find missing sides or angles in triangles.

Think of trigonometry as a tool that helps us measure things we cannot reach directly. For example, how tall is a tree? How far is a mountain? Trigonometry can help us find these answers using triangles.

The right triangle

Before we learn trigonometry, we need to understand right triangles. A right triangle is a triangle with one corner that is exactly 90 degrees (a square corner).

Parts of a Right Triangle:

The Right Angle: This is the square corner (90 degrees). It looks like the corner of a book or a square.

The Hypotenuse: This is the longest side of the triangle. It is always opposite (across from) the right angle. Think of it as the “hero” side because it’s the biggest.

The Two Other Sides: These are called legs. They meet at the right angle to form the square corner.

Important: In trigonometry, we give special names to the sides based on which angle we are looking at.

The three main trigonometry rules

There are three main rules in trigonometry. They help us find missing parts of right triangles. These rules are called sine, cosine, and tangent.

Sine (sin)

What it means: Sine compares two sides of a right triangle.

The Rule: Sine = Opposite side ÷ Hypotenuse

How to remember: “Sine is Opposite over Hypotenuse”

When to use: When you know an angle and want to find the side opposite to that angle, or when you know two sides and want to find an angle.

Cosine (cos)

What it means: Cosine also compares two sides of a right triangle.

The Rule: Cosine = Adjacent side ÷ Hypotenuse

How to remember: “Cosine is Adjacent over Hypotenuse”

When to use: When you know an angle and want to find the side next to that angle, or when you know two sides and want to find an angle.

Tangent (tan)

What it means: Tangent compares the two shorter sides of a right triangle.

The Rule: Tangent = Opposite side ÷ Adjacent side

How to remember: “Tangent is Opposite over Adjacent”

When to use: When you want to compare the two sides that are not the hypotenuse.

Easy way to remember: soh-cah-toa

This is a simple way to remember all three rules:

S-O-H: Sine = Opposite ÷ Hypotenuse C-A-H: Cosine = Adjacent ÷ Hypotenuse
T-O-A: Tangent = Opposite ÷ Adjacent

You can remember this by saying “SOH-CAH-TOA” like a chant or thinking of it as “Some Old Hippie Caught Another Hippie Tripping On Acid” (though any silly sentence that helps you remember works!).

Understanding opposite and adjacent

This is often the hardest part for students. Let’s make it simple:

Step 1: Find the right angle (90 degrees) Step 2: Pick one of the other two angles to work with Step 3: From that angle:

• Opposite side: The side across from your chosen angle (not touching it)
• Adjacent side: The side next to your chosen angle (touching it, but not the hypotenuse)
• Hypotenuse: Always the longest side (opposite the right angle)

Memory Trick: Think of “adjacent” like “next to” – the adjacent side is next to your angle.

Simple examples

Example 1: Finding a Side You have a right triangle. One angle is 30 degrees. The hypotenuse is 10 units long. You want to find the side opposite the 30-degree angle.

Solution:

• Use sine because you have hypotenuse and want opposite
• Sin(30°) = Opposite ÷ 10
• Sin(30°) = 0.5 (this is a known value)
• 0.5 = Opposite ÷ 10
• Opposite = 0.5 × 10 = 5 units

Example 2: Finding an Angle You have a right triangle. The opposite side is 3 units and the adjacent side is 4 units. You want to find the angle.

Solution:

• Use tangent because you have opposite and adjacent
• Tan(angle) = 3 ÷ 4 = 0.75
• Angle = arctan(0.75) ≈ 37 degrees

Real-life uses of trigonometry

Trigonometry is not just for math class. It’s used in many real situations:

Building and Construction:

• Making sure roofs have the right slope
• Building ramps for wheelchairs
• Constructing stairs at safe angles

Navigation:

• Ships use trigonometry to find their way
• Airplanes use it to land safely
• GPS systems use trigonometric calculations

Measuring Heights:

• Finding how tall buildings are without climbing them
• Measuring the height of trees or mountains
• Determining how deep wells or holes are

Entertainment:

• Video games use trigonometry for graphics
• Movies use it for special effects
• Music uses trigonometric waves for sound

Sports:

• Basketball players use angles to make shots
• Golfers use trigonometry to plan their shots
• Engineers design sports equipment using these principles

Common angles and their values

Some angles appear often in trigonometry. It’s helpful to know their values:

30 degrees:

• Sin(30°) = 0.5
• Cos(30°) = 0.866
• Tan(30°) = 0.577

45 degrees:

• Sin(45°) = 0.707
• Cos(45°) = 0.707
• Tan(45°) = 1

60 degrees:

• Sin(60°) = 0.866
• Cos(60°) = 0.5
• Tan(60°) = 1.732

USING A CALCULATOR

Most calculators have SIN, COS, and TAN buttons. Here’s how to use them:

To find sine, cosine, or tangent of an angle:

1. Make sure your calculator is in “degree” mode (not radians)
2. Type the angle number
3. Press SIN, COS, or TAN
4. Read the answer

To find an angle when you know the ratio:

1. Calculate the ratio (opposite ÷ hypotenuse, for example)
2. Press the inverse button (usually marked as SIN⁻¹, COS⁻¹, or TAN⁻¹)
3. The calculator will show the angle

Step-by-step problem solving

Step 1: Draw the triangle and label what you know Step 2: Identify what you’re trying to find (side or angle) Step 3: Choose which trigonometric rule to use (SOH-CAH-TOA) Step 4: Set up the equation Step 5: Solve using a calculator Step 6: Check if your answer makes sense

Practice problems

Problem 1: In a right triangle, one angle is 40 degrees and the hypotenuse is 15 units. Find the opposite side.

Problem 2: The opposite side is 8 units and the adjacent side is 6 units. Find the angle.

Problem 3: A ladder leans against a wall at 60 degrees. The ladder is 12 feet long. How high up the wall does it reach?

Common mistakes to avoid

• Forgetting to check if your calculator is in degree mode
• Mixing up opposite and adjacent sides
• Using the wrong trigonometric function
• Forgetting that the hypotenuse is always the longest side
• Not drawing a diagram to visualize the problem

SUMMARY

Trigonometry helps us work with right triangles when we don’t know all the measurements. The three main rules are:

• Sine: opposite ÷ hypotenuse
• Cosine: adjacent ÷ hypotenuse
• Tangent: opposite ÷ adjacent

Remember SOH-CAH-TOA, and always draw a picture to help you see which sides are opposite, adjacent, and hypotenuse.

Trigonometry is everywhere around us – from the buildings we live in to the games we play. Once you understand these basics, you’ll start seeing triangles and angles in a whole new way!

HOMEWORK

1. Practice identifying opposite, adjacent, and hypotenuse in 5 different right triangles
2. Use a calculator to find sin(45°), cos(30°), and tan(60°)
3. Solve: In a right triangle with a 50° angle and hypotenuse of 20 units, find the opposite side
4. Look around your home and find 3 examples where trigonometry might be useful