Trigonometry – JSS3 Mathematics Lesson Note (Second Term)

Trigonometry is the branch of mathematics that uses the sides and angles of right triangles to find missing lengths and angles.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Explain what trigonometry means.
2. Name the parts of a right triangle.
3. State the three main trigonometric ratios (sine, cosine and tangent).
4. Use simple trigonometry to find a missing side or angle.
5. Give examples of trigonometry in everyday life.

What is trigonometry?

Trigonometry is the study of the relationship between the sides and angles of triangles.

• Meaning of the word: it comes from Greek words meaning “triangle” and “measure”.
• Main use: it finds missing sides or angles in right triangles.
• Why it matters: it measures things we cannot reach directly, such as the height of a tree or a tall building.
• Tools used: three ratios called sine, cosine and tangent.

What is a right triangle?

A right triangle is a triangle with one angle of exactly 90 degrees (a square corner).

Parts of a right triangle

• Right angle: the 90-degree corner, like the corner of a book.
• Hypotenuse: the longest side, always opposite the right angle.
• Legs: the two shorter sides that meet at the right angle.

• Note: in trigonometry, the two legs are named “opposite” and “adjacent” depending on the angle you are using.

How do you identify the opposite and adjacent sides?

The opposite and adjacent sides are named from the angle you have chosen, and they change if you choose the other angle.

1. Find the right angle (90 degrees).
2. Pick one of the other two angles to work with.
3. Find the hypotenuse: the longest side, opposite the right angle.
4. Find the opposite side: the side across from your chosen angle, not touching it.
5. Find the adjacent side: the side next to your chosen angle that is not the hypotenuse.

• Memory trick: “adjacent” means “next to”, so the adjacent side is next to your angle.

What are the three trigonometric ratios?

There are three main trigonometric ratios: sine, cosine and tangent.

What is sine (sin)?

Sine of an angle is the opposite side divided by the hypotenuse.

• Rule: sin θ = Opposite ÷ Hypotenuse.
• Use it when: the sides involved are the opposite and the hypotenuse.

What is cosine (cos)?

Cosine of an angle is the adjacent side divided by the hypotenuse.

• Rule: cos θ = Adjacent ÷ Hypotenuse.
• Use it when: the sides involved are the adjacent and the hypotenuse.

What is tangent (tan)?

Tangent of an angle is the opposite side divided by the adjacent side.

• Rule: tan θ = Opposite ÷ Adjacent.
• Use it when: the sides involved are the two legs, and the hypotenuse is not used.

What is SOH-CAH-TOA?

SOH-CAH-TOA is a memory aid for the three ratios.

• SOH: Sine = Opposite ÷ Hypotenuse.
• CAH: Cosine = Adjacent ÷ Hypotenuse.
• TOA: Tangent = Opposite ÷ Adjacent.

How do you rearrange the ratios to find a side?

• Opposite: Opposite = Hypotenuse × sin θ.
• Adjacent: Adjacent = Hypotenuse × cos θ.
• Opposite from adjacent: Opposite = Adjacent × tan θ.

What are some worked examples?

Example 1: Finding a side with sine

• Question: one angle is 30° and the hypotenuse is 10 units. Find the side opposite the 30° angle.

1. Use sine, because the opposite and the hypotenuse are involved.
2. sin 30° = Opposite ÷ 10.
3. sin 30° = 0.5 (from the table of common angles).
4. Opposite = 0.5 × 10 = 5 units.

Example 2: Finding a side with cosine

• Question: one angle is 60° and the hypotenuse is 8 units. Find the adjacent side.

1. Use cosine, because the adjacent and the hypotenuse are involved.
2. cos 60° = Adjacent ÷ 8.
3. Adjacent = 8 × 0.5 = 4 units.

Example 3: Finding an angle with tangent

• Question: the opposite side is 3 units and the adjacent side is 4 units. Find the angle.

1. Use tangent, because the opposite and the adjacent are involved.
2. tan θ = 3 ÷ 4 = 0.75.
3. θ = tan⁻¹(0.75) ≈ 37°.

What are the values of common angles?

The sine, cosine and tangent of 30°, 45° and 60° appear often, so they are worth learning.

• Note: the values are rounded to three decimal places.

How do you use a calculator for trigonometry?

Most scientific calculators have SIN, COS and TAN buttons for finding ratios and angles.

How do you find the sine, cosine or tangent of an angle?

1. Set the calculator to degree mode (not radians).
2. Type the angle.
3. Press SIN, COS or TAN.
4. Read the answer.

• Note: on some calculators, you press SIN, COS or TAN first and then type the angle.

How do you find an angle from a ratio?

1. Work out the ratio (for example, opposite ÷ hypotenuse).
2. Press the inverse button (SIN⁻¹, COS⁻¹ or TAN⁻¹), usually after pressing SHIFT or 2nd.
3. Enter the ratio and read the angle.

How do you solve a trigonometry problem step by step?

1. Draw the triangle and label what you know.
2. Decide what you need to find (a side or an angle).
3. Name the opposite, adjacent and hypotenuse from the chosen angle.
4. Choose the ratio using SOH-CAH-TOA.
5. Write the equation and rearrange it.
6. Solve with a calculator.
7. Check the answer: the hypotenuse must be the longest side.

Where is trigonometry used in real life?

Trigonometry is used in many jobs and activities outside the mathematics class.

Building and construction

• Roofs: giving a roof the right slope.
• Ramps: building ramps for wheelchairs.
• Stairs: building stairs at safe angles.

Navigation

• Ships: finding direction and position at sea.
• Aeroplanes: planning the angle of descent for landing.
• GPS: using trigonometric calculations to fix a position.

Measuring heights and depths

• Buildings: finding height without climbing.
• Trees and mountains: finding height from a distance.
• Wells and holes: finding depth.

Entertainment

• Video games: producing graphics and movement.
• Films: creating special effects.
• Music: describing sound waves with sine waves.

Sports

• Basketball and golf: the angle of a shot can be studied with trigonometry.
• Equipment: engineers use these ideas to design sports equipment.

What mistakes should you avoid?

• Wrong calculator mode: forgetting to set degree mode.
• Mixed-up sides: confusing opposite and adjacent.
• Wrong ratio: using sine when cosine or tangent is needed.
• Hypotenuse error: forgetting that it is always the longest side.
• No diagram: skipping the sketch of the triangle.

Evaluation

1. State the three trigonometric ratios using SOH-CAH-TOA.
2. In a right triangle, one angle is 40° and the hypotenuse is 15 units. Find the opposite side.
3. The opposite side is 8 units and the adjacent side is 6 units. Find the angle.
4. A ladder 12 metres long leans against a wall at 60° to the ground. How high up the wall does it reach?

Assignment

1. Draw a right triangle with a 50° angle and a hypotenuse of 20 units. Label the opposite, adjacent and hypotenuse, then find the opposite side.