DEFINITION

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right-angled triangles.

It helps us to find unknown sides or angles of triangles using simple ratios.

In simple words: Trigonometry is like a tool that tells us how the sides of a triangle relate to its angles.

KEY TERMS IN TRIGONOMETRY

Right-angled triangle: A triangle that has one angle equal to 90° (a right angle).

Hypotenuse: The longest side of a right-angled triangle, always opposite the right angle.

Opposite side: The side that is opposite the angle you are considering.

Adjacent side: The side that is next to the angle you are considering, but not the hypotenuse.

TRIGONOMETRIC RATIOS

For a right-angled triangle with an angle θ:

Sine (sin θ) = Opposite ÷ Hypotenuse

Cosine (cos θ) = Adjacent ÷ Hypotenuse

Tangent (tan θ) = Opposite ÷ Adjacent

Note: These ratios only work for right-angled triangles.

EXPLANATION

Imagine a triangle. If you know one angle (other than the right angle) and one side, you can find the other sides using sine, cosine, or tangent.

These ratios are fixed for each angle. For example, sin 30° is always 0.5.

        |\
        | \
        |  \
Opposite|   \ Hypotenus
        |    \
        |     \
        |______\
        Adjacent
    

θ is the angle we are considering.

The side opposite θ is called the opposite.

The side next to θ (but not the hypotenuse) is called the adjacent.

The longest side is the hypotenuse.

STEPS TO USE TRIGONOMETRY TO FIND SIDES OR ANGLES

1. Identify the given angle and sides.


2. Decide which trigonometric ratio to use (sine, cosine, or tangent).


3. Write the formula for the ratio.


4. Substitute the known values.


5. Solve for the unknown side or angle.

EXAMPLES

Example 1 – Find a side using sine

A right-angled triangle has an angle 30° and a hypotenuse of 10 cm. Find the length of the side opposite the angle.

Solution:

sin θ = Opposite ÷ Hypotenuse

sin 30° = Opposite ÷ 10

0.5 = Opposite ÷ 10

Opposite = 0.5 × 10 = 5 cm

Example 2 – Find a side using cosine

A right-angled triangle has an angle 60° and a hypotenuse of 12 cm. Find the length of the adjacent side.

Solution:

cos θ = Adjacent ÷ Hypotenuse

cos 60° = Adjacent ÷ 12

0.5 = Adjacent ÷ 12

Adjacent = 0.5 × 12 = 6 cm

Example 3 – Find a side using tangent

A right-angled triangle has an angle 45° and an adjacent side of 7 cm. Find the length of the opposite side.

Solution:

tan θ = Opposite ÷ Adjacent

tan 45° = Opposite ÷ 7

1 = Opposite ÷ 7

Opposite = 1 × 7 = 7 cm

Example 4 – Find an angle

A right-angled triangle has an opposite side of 3 cm and adjacent side of 4 cm. Find the angle θ.

Solution:

tan θ = Opposite ÷ Adjacent

tan θ = 3 ÷ 4

θ = tan⁻¹ (0.75) ≈ 37°

REAL-LIFE APPLICATIONS OF TRIGONOMETRY

• Measuring heights of tall buildings or trees.


• Finding distances across rivers or lakes without crossing them.


• Designing ramps, roofs, and slopes in construction.


• Navigation and map reading.

TIPS TO REMEMBER

• Always label the triangle with the angle you are considering.


• Hypotenuse is always opposite the right angle.


• The side opposite the angle of interest is “opposite,” and the other side next to the angle is “adjacent.”


• Use a calculator for sine, cosine, and tangent values.

SUMMARY

• Trigonometry studies the relationship between sides and angles in triangles.


• The main ratios are sine, cosine, and tangent.


• These ratios help find unknown sides or angles in right-angled triangles.


• Trigonometry has practical applications in construction, navigation, and measurement.

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DEFINITION OF TRIGONOMETRY

Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of triangles, especially right-angled triangles.

It helps us to find unknown sides or angles of triangles using special ratios.

In simple words: Trigonometry is like a tool that helps us to know how the sides of a triangle relate to its angles, without measuring every side directly.

KEY TERMS

Right-angled triangle: A triangle that has one angle equal to 90°. This is called the right angle.

Hypotenuse: The longest side of a right-angled triangle. It is always opposite the right angle.

Opposite side: The side opposite the angle we are looking at.

Adjacent side: The side next to the angle we are looking at, but not the hypotenuse.

THE SINE, COSINE, AND TANGENT RATIOS

For a right-angled triangle, if we are considering an acute angle θ (any angle less than 90°):

Sine (sin θ)

Definition: Sine is the ratio of the opposite side to the hypotenuse.

Formula: sin θ = Opposite side ÷ Hypotenuse

Example: If the opposite side is 3 cm and the hypotenuse is 5 cm, then sin θ = 3 ÷ 5 = 0.6

Cosine (cos θ)

Definition: Cosine is the ratio of the adjacent side to the hypotenuse.

Formula: cos θ = Adjacent side ÷ Hypotenuse

Example: If the adjacent side is 4 cm and the hypotenuse is 5 cm, then cos θ = 4 ÷ 5 = 0.8

Tangent (tan θ)

Definition: Tangent is the ratio of the opposite side to the adjacent side.

Formula: tan θ = Opposite side ÷ Adjacent side

Example: If the opposite side is 3 cm and the adjacent side is 4 cm, then tan θ = 3 ÷ 4 = 0.75

STEP-BY-STEP GUIDE TO USING SINE, COSINE, AND TANGENT

Step 1 – Identify the angle of interest: Choose the angle θ you want to work with. This is usually one of the acute angles in a right-angled triangle.

Step 2 – Label the sides:

Hypotenuse: Always opposite the right angle.

Opposite: Opposite to angle θ.

Adjacent: Next to angle θ, not the hypotenuse.

Step 3 – Decide which ratio to use:

If you know the opposite and hypotenuse → use sine.

If you know the adjacent and hypotenuse → use cosine.

If you know the opposite and adjacent → use tangent.

Step 4 – Write the ratio formula: Example: sin θ = Opposite ÷ Hypotenuse

Step 5 – Substitute the known values: Plug in the numbers you know into the formula.

Step 6 – Solve for the unknown side or angle: For sides: Multiply or divide to isolate the unknown. For angles: Use the inverse function on a calculator, e.g., θ = sin⁻¹(Opposite ÷ Hypotenuse).

EXAMPLES WITH STEP-BY-STEP SOLUTIONS

Example 1 – Find the opposite side using sine

A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the opposite side.

Solution:

Step 1: Identify the angle: θ = 30°

Step 2: Decide the ratio: Opposite side is unknown, hypotenuse is known → use sine

Step 3: Write the formula: sin θ = Opposite ÷ Hypotenuse

Step 4: Substitute: sin 30° = Opposite ÷ 10

Step 5: Solve: 0.5 = Opposite ÷ 10 → Opposite = 0.5 × 10 = 5 cm

Opposite side = 5 cm

Example 2 – Find the adjacent side using cosine

A right-angled triangle has a hypotenuse of 12 cm and an angle of 60°. Find the adjacent side.

Solution:

Step 1: Identify the angle: θ = 60°

Step 2: Decide the ratio: Adjacent side is unknown, hypotenuse is known → use cosine

Step 3: Write the formula: cos θ = Adjacent ÷ Hypotenuse

Step 4: Substitute: cos 60° = Adjacent ÷ 12

Step 5: Solve: 0.5 = Adjacent ÷ 12 → Adjacent = 0.5 × 12 = 6 cm

Adjacent side = 6 cm

Example 3 – Find the opposite side using tangent

A right-angled triangle has an angle of 45° and an adjacent side of 7 cm. Find the opposite side.

Solution:

Step 1: Identify the angle: θ = 45°

Step 2: Decide the ratio: Opposite side is unknown, adjacent is known → use tangent

Step 3: Write the formula: tan θ = Opposite ÷ Adjacent

Step 4: Substitute: tan 45° = Opposite ÷ 7

Step 5: Solve: 1 = Opposite ÷ 7 → Opposite = 1 × 7 = 7 cm

Opposite side = 7 cm

Example 4 – Find an angle using tangent

A right-angled triangle has an opposite side of 3 cm and adjacent side of 4 cm. Find angle θ.

Solution:

Step 1: Decide the ratio: Angle unknown, opposite and adjacent known → use tangent

Step 2: Write the formula: tan θ = Opposite ÷ Adjacent

Step 3: Substitute: tan θ = 3 ÷ 4 = 0.75

Step 4: Solve: θ = tan⁻¹(0.75) ≈ 37°

Angle θ ≈ 37°

SUMMARY AND KEY POINTS

• Trigonometry studies the relationship between sides and angles of right-angled triangles.


• The main ratios are:


• Sine = Opposite ÷ Hypotenuse


• Cosine = Adjacent ÷ Hypotenuse


• Tangent = Opposite ÷ Adjacent


• These ratios help us to find unknown sides or angles in right-angled triangles.


• Steps to solve: Identify angle → label sides → choose ratio → substitute → solve.


• Trigonometry has real-life applications: measuring heights of buildings, designing ramps, navigation, and construction.

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DIAGRAM OF A RIGHT-ANGLED TRIANGLE

              /|
             / |
            /  |
           /   |  ← Opposite side (to angle θ)
          /θ___|
         ← Adjacent side
   (next to angle θ, not the longest)

Hypotenuse = the longest side (always opposite the right angle)

Step-by-Step Explanation of the Diagram

Look at the triangle above carefully. It has a small square corner (that means a 90° angle) — that is why it is called a right-angled triangle.

Angle θ (theta): This is the small angle near the bottom left of the triangle.

The Hypotenuse

This is the longest side of the triangle. It is always across from the right angle (90°). No matter which angle θ you choose, the hypotenuse does not change.

The Opposite side

This is the side that is across from the angle θ you are working with. It is called "opposite" because it is facing the angle.

The Adjacent side

This is the side that is next to (or beside) the angle θ. It is not the hypotenuse. It touches the angle θ directly.

How to Remember Them Easily

You can remember the three trigonometric ratios using a simple memory trick: “SOH CAH TOA”

Ratio	Formula	Meaning
Sine	sin θ = Opposite ÷ Hypotenuse	S = Sine, O = Opposite, H = Hypotenuse
Cosine	cos θ = Adjacent ÷ Hypotenuse	C = Cosine, A = Adjacent, H = Hypotenuse
Tangent	tan θ = Opposite ÷ Adjacent	T = Tangent, O = Opposite, A = Adjacent

So:

SOH → sin θ = Opposite / Hypotenuse

CAH → cos θ = Adjacent / Hypotenuse

TOA → tan θ = Opposite / Adjacent

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DEFINITION

Trigonometric ratios (sine, cosine, and tangent) are special ratios in a right-angled triangle that help us to find unknown sides or angles.

In simple words: Trigonometric ratios help us to solve real-life problems where we need to know distances or heights without measuring them directly, just by knowing an angle and one side of a triangle.

KEY TERMS REMINDER

• Right-angled triangle: A triangle with one 90° angle.


• Hypotenuse: The longest side, opposite the right angle.


• Opposite side: The side across from the angle of interest.


• Adjacent side: The side next to the angle of interest (not the hypotenuse).

STEPS TO SOLVE PROBLEMS USING TRIGONOMETRIC RATIOS

1. Identify the triangle and mark the known angle and sides.


2. Label the sides as Opposite, Adjacent, and Hypotenuse with respect to the angle you are working with.


3. Choose the correct ratio (sine, cosine, tangent) based on which sides you know and which you want to find.


4. Write the formula for the chosen ratio.


5. Substitute the known values into the formula.


6. Solve for the unknown side or angle.

EXAMPLES WITH STEP-BY-STEP SOLUTIONS

Example 1 – Find the height of a tree

A tree casts a shadow of 8 m. The angle of elevation of the sun is 30°. Find the height of the tree.

Solution:

1. Identify the angle: θ = 30° (angle of elevation).


2. Label sides: Opposite = height of tree (unknown), Adjacent = 8 m (shadow), Hypotenuse = not given.


3. Choose ratio: Opposite and Adjacent → use tangent.


4. Write formula: tan θ = Opposite ÷ Adjacent


5. Substitute: tan 30° = Opposite ÷ 8


6. Solve: 0.577 = Opposite ÷ 8 → Opposite = 0.577 × 8 ≈ 4.62 m

Height of the tree ≈ 4.62 m

Example 2 – Find the length of a ladder

A ladder leans against a wall making an angle of 60° with the ground. The foot of the ladder is 3 m away from the wall. Find the length of the ladder.

Solution:

1. Identify the angle: θ = 60°.
2. Label sides: Adjacent = 3 m (distance from wall), Hypotenuse = ladder length (unknown), Opposite = height on wall.


3. Choose ratio: Adjacent and Hypotenuse → use cosine.


4. Write formula: cos θ = Adjacent ÷ Hypotenuse


5. Substitute: cos 60° = 3 ÷ Hypotenuse


6. Solve: 0.5 = 3 ÷ Hypotenuse → Hypotenuse = 3 ÷ 0.5 = 6 m

Length of the ladder = 6 m

Example 3 – Find the distance across a river

A person stands on one bank of a river. The angle of elevation to a tree on the other bank is 40°. The height of the tree is 10 m. Find the distance across the river (base of the tree to the person).

Solution:

1. Identify the angle: θ = 40°.


2. Label sides: Opposite = 10 m (height of tree), Adjacent = distance across the river (unknown), Hypotenuse = not needed.


3. Choose ratio: Opposite and Adjacent → use tangent.


4. Write formula: tan θ = Opposite ÷ Adjacent


5. Substitute: tan 40° = 10 ÷ Adjacent


6. Solve: 0.8391 = 10 ÷ Adjacent → Adjacent = 10 ÷ 0.8391 ≈ 11.92 m

Distance across the river ≈ 11.92 m

Example 4 – Find the angle of elevation

A building is 15 m high. A person stands 20 m from the base. Find the angle of elevation to the top of the building.

Solution:

1. Identify the sides: Opposite = 15 m, Adjacent = 20 m, Hypotenuse = not needed.


2. Choose ratio: Opposite and Adjacent → use tangent.


3. Write formula: tan θ = Opposite ÷ Adjacent


4. Substitute: tan θ = 15 ÷ 20 = 0.75


5. Solve: θ = tan⁻¹(0.75) ≈ 37°

Angle of elevation ≈ 37°

Example 5 – Find the height of a ramp

A ramp rises at an angle of 25° to the ground. The ramp is 12 m long. Find the height it rises to.

Solution:

1. Identify the angle: θ = 25°.
2. Label sides: Hypotenuse = 12 m, Opposite = height (unknown), Adjacent = base (not needed).


3. Choose ratio: Opposite and Hypotenuse → use sine.


4. Write formula: sin θ = Opposite ÷ Hypotenuse


5. Substitute: sin 25° = Opposite ÷ 12


6. Solve: 0.4226 = Opposite ÷ 12 → Opposite = 0.4226 × 12 ≈ 5.07 m

Height of the ramp ≈ 5.07 m

SUMMARY

• Identify the angle of interest and label sides correctly.


• Choose the correct trigonometric ratio: sine, cosine, or tangent.


• Use the ratio formula and substitute known values.


• Solve step by step for the unknown side or angle.


• Trigonometry is widely used in real-life: measuring heights, distances, building ramps, and navigation.

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Trigonometry Questions for JSSCE (with Step-by-Step Solutions)

Question 1

A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the length of the side opposite the angle.

Solution:

1. Identify the angle: θ = 30°
2. Known side: Hypotenuse = 10 cm; unknown side = Opposite → use sine.
3. Formula: sin θ = Opposite ÷ Hypotenuse
4. Substitute: sin 30° = Opposite ÷ 10 → 0.5 = Opposite ÷ 10
5. Solve: Opposite = 0.5 × 10 = 5 cm

Answer: Opposite side = 5 cm

Question 2

A ladder leans against a wall at an angle of 60° with the ground. The foot of the ladder is 3 m from the wall. Find the length of the ladder.

Solution:

1. Label sides: Adjacent = 3 m, Hypotenuse = ladder length (unknown), angle θ = 60°.
2. Use cosine: cos θ = Adjacent ÷ Hypotenuse
3. Substitute: cos 60° = 3 ÷ Hypotenuse → 0.5 = 3 ÷ Hypotenuse
4. Solve: Hypotenuse = 3 ÷ 0.5 = 6 m

Answer: Ladder length = 6 m

Question 3

A tree casts a shadow 8 m long. The angle of elevation of the sun is 30°. Find the height of the tree.

Solution:

1. Label sides: Opposite = height of tree (unknown), Adjacent = shadow = 8 m, angle θ = 30°.
2. Use tangent: tan θ = Opposite ÷ Adjacent
3. Substitute: tan 30° = Opposite ÷ 8 → 0.577 = Opposite ÷ 8
4. Solve: Opposite = 0.577 × 8 ≈ 4.62 m

Answer: Height ≈ 4.62 m

Question 4

A ramp rises at an angle of 25° and is 12 m long. Find the vertical height it rises to.

Solution:

1. Label sides: Hypotenuse = 12 m, Opposite = height (unknown), angle θ = 25°.
2. Use sine: sin θ = Opposite ÷ Hypotenuse
3. Substitute: sin 25° = Opposite ÷ 12 → 0.4226 = Opposite ÷ 12
4. Solve: Opposite = 0.4226 × 12 ≈ 5.07 m

Answer: Height ≈ 5.07 m

Question 5

A right-angled triangle has an opposite side of 4 cm and an adjacent side of 3 cm. Find the angle opposite the 4 cm side.

Solution:

1. Use tangent: tan θ = Opposite ÷ Adjacent = 4 ÷ 3 = 1.3333
2. θ = tan⁻¹(1.3333) ≈ 53°

Answer: Angle ≈ 53°

Question 6

A right-angled triangle has an angle of 45° and a hypotenuse of 7 cm. Find the length of the side adjacent to the angle.

Solution:

1. Use cosine: cos θ = Adjacent ÷ Hypotenuse
2. Substitute: cos 45° = Adjacent ÷ 7 → 0.707 = Adjacent ÷ 7
3. Solve: Adjacent = 0.707 × 7 ≈ 4.95 cm

Answer: Adjacent ≈ 4.95 cm

Question 7

A building is 20 m tall. A person is 15 m from its base. Find the angle of elevation to the top of the building.

Solution:

1. Label sides: Opposite = 20 m, Adjacent = 15 m, angle θ = unknown
2. Use tangent: tan θ = Opposite ÷ Adjacent = 20 ÷ 15 = 1.3333
3. θ = tan⁻¹(1.3333) ≈ 53°

Answer: Angle of elevation ≈ 53°

Question 8

A ramp is 10 m long and rises 6 m. Find the angle the ramp makes with the horizontal.

Solution:

1. Label sides: Opposite = 6 m, Hypotenuse = 10 m, angle θ = unknown
2. Use sine: sin θ = Opposite ÷ Hypotenuse = 6 ÷ 10 = 0.6
3. θ = sin⁻¹(0.6) ≈ 37°

Answer: Angle ≈ 37°

Question 9

A ladder reaches a window 9 m high. The ladder makes an angle of 60° with the ground. Find the length of the ladder.

Solution:

1. Label sides: Opposite = 9 m, Hypotenuse = ladder (unknown), angle θ = 60°
2. Use sine: sin θ = Opposite ÷ Hypotenuse
3. Substitute: sin 60° = 9 ÷ Hypotenuse → 0.866 = 9 ÷ Hypotenuse
4. Solve: Hypotenuse = 9 ÷ 0.866 ≈ 10.39 m

Answer: Ladder length ≈ 10.39 m

Question 10

A right-angled triangle has sides 5 cm (opposite) and 12 cm (adjacent). Find the angle opposite the 5 cm side and the hypotenuse.

Solution:

1. Find the angle using tangent: tan θ = Opposite ÷ Adjacent = 5 ÷ 12 ≈ 0.4167
2. θ = tan⁻¹(0.4167) ≈ 23°
3. Find hypotenuse using Pythagoras theorem: Hypotenuse² = 5² + 12² = 25 + 144 = 169 → Hypotenuse = √169 = 13 cm

Answer: Angle ≈ 23°, Hypotenuse = 13 cm