Trigonometry

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Notes

Trigonometry Basics

Trigonometry is the mathematics of angles in triangles, studying relationships between side lengths and angles.
The three trigonometric functions are **sine**, **cosine**, and **tangent** (sin, cos, tan).
These functions are ratios of side lengths in **right-angled triangles**.
SOHCAHTOA is a mnemonic to remember the ratios: **S**in = **O**pposite/**H**ypotenuse, **C**os = **A**djacent/**H**ypotenuse, **T**an = **O**pposite/**A**djacent.
Trigonometry (like Pythagoras) can only be used in **right-angled triangles**.
Ensure your calculator is set to **degrees** (D or Deg on screen).

Labelling Sides of a Right-Angled Triangle

Label the sides relative to a chosen angle θ: **Hypotenuse (H)** – longest side, opposite the right angle.
**Opposite (O)** – side directly opposite angle θ.
**Adjacent (A)** – side next to angle θ (not the hypotenuse).
H is always the same; O and A change depending on which angle is θ.

Finding Missing Lengths Using SOHCAHTOA

**Step 1:** Label the sides as H, O, A relative to the given angle.
**Step 2:** Identify which ratio to use: given two sides, find the two letters in SOHCAHTOA (e.g., O and A → tan).
**Step 3:** Substitute values into the formula: e.g., tan(θ) $= \frac{O}{A}$ .
**Step 4:** Rearrange to solve for the unknown side (multiply or divide).
**Step 5:** Type into calculator and round to **3 significant figures** unless otherwise stated.
Example: If $\tan (43^{\circ}) = \frac{x}{9}$ , then $x = 9 \times \tan (43^{\circ}) = 8.39$ cm (3 s.f.).

Finding Missing Angles Using SOHCAHTOA

**Step 1:** Label sides as H, O, A relative to the unknown angle.
**Step 2:** Identify the ratio using the two given side letters (e.g., A and H → cos).
**Step 3:** Write the ratio as a fraction: cos(θ) $= \frac{A}{H}$ .
**Step 4:** Use the **inverse trigonometric function** (sin⁻¹, cos⁻¹, tan⁻¹) – press SHIFT on calculator.
**Step 5:** Calculate and round to **1 decimal place** unless otherwise stated.
Example: If $\cos (y) = \frac{8}{23}$ , then $y =$ cos⁻¹(8/23) $= 69.6^{\circ} (1 d.p.)$ .

Common Exam Tips

Always label the triangle first before applying SOHCAHTOA.
Write down the ratio you are using to avoid mistakes.
For lengths, round to **3 significant figures**; for angles, round to **1 decimal place**.
Check your calculator mode: it must be in **degrees**.
SOHCAHTOA only works in **right-angled triangles** – do not use it for non-right triangles.

Labelling Sides of a Right-Angled Triangle

SOHCAHTOA Triangle

Finding a Length Example

Finding an Angle Example

Practice questions

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1.In a right-angled triangle, which side is the hypotenuse?
Easy
- AThe side opposite the right angle
- BThe side opposite the given angle
- CThe side adjacent to the given angle
- DThe shortest side
2.What does SOHCAHTOA stand for?
Easy
- A $Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent$
- B $Sin = Adjacent/Hypotenuse, Cos = Opposite/Hypotenuse, Tan = Adjacent/Opposite$
- C $Sin = Hypotenuse/Opposite, Cos = Hypotenuse/Adjacent, Tan = Adjacent/Opposite$
- D $Sin = Opposite/Adjacent, Cos = Adjacent/Hypotenuse, Tan = Hypotenuse/Opposite$
3.If tan θ $= opposite/adjacent$ , which sides are used for tan?
Easy
- AOpposite and adjacent
- BOpposite and hypotenuse
- CAdjacent and hypotenuse
- DHypotenuse and opposite
4.In a right-angled triangle, the side opposite the angle θ is 5 cm and the hypotenuse is 13 cm. What is sin θ?
Easy
- A $\frac{5}{13}$
- B $\frac{13}{5}$
- C $\frac{12}{13}$
- D $\frac{5}{12}$
5.In a right-angled triangle, the adjacent side is 8 cm and the hypotenuse is 17 cm. What is cos θ?
Easy
- A $\frac{8}{17}$
- B $\frac{17}{8}$
- C $\frac{15}{17}$
- D $\frac{8}{15}$
6.A right-angled triangle has an angle of 30° and an adjacent side of 10 cm. Find the length of the opposite side.
Medium
- A5.77 cm
- B5.00 cm
- C8.66 cm
- D11.55 cm
7.In a right-angled triangle, the opposite side is 12 cm and the hypotenuse is 13 cm. Find the angle θ.
Medium
- A67.4°
- B22.6°
- C45.0°
- D60.0°
8.A right-angled triangle has an angle of 40° and an opposite side of 7 cm. Find the hypotenuse.
Medium
- A10.9 cm
- B9.1 cm
- C8.4 cm
- D11.5 cm

Summary

## Trigonometry Revision Notes ### SOHCAHTOA - **S**in = **O**pposite / **H**ypotenuse - **C**os = **A**djacent / **H**ypotenuse - **T**an = **O**pposite / **A**djacent ### Finding Lengths 1. Label sides: H (hypotenuse, opposite right angle), O (opposite given angle), A (adjacent to given angle). 2. Choose ratio: given two letters (e.g., O and A → tan). 3. Write formula: e.g., tan(θ) = O/A. 4. Substitute and solve: multiply or divide. 5. Round to 3 significant figures unless specified. ### Finding Angles 1. Label sides as above. 2. Choose ratio using given sides. 3. Write formula with unknown angle: e.g., cos(θ) = A/H. 4. Use inverse function: θ = cos⁻¹(A/H). 5. Round to 1 decimal place. ### Key Points - Only for right-angled triangles. - Calculator must be in degree mode (D or Deg). - Hypotenuse is always opposite the right angle. ### Example (Length) Find x if tan(43°) = x/9 → x = 9 × tan(43°) = 8.39 cm (3 s.f.) ### Example (Angle) Find y if cos(y) = 8/23 → y = cos⁻¹(8/23) = 69.6° (1 d.p.)

Key terms

Hypotenuse: The longest side in a right-angled triangle, opposite the right angle.
Opposite: The side opposite the chosen angle θ.
Adjacent: The side next to the chosen angle θ.
SOHCAHTOA: A mnemonic for remembering trigonometric ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
sin θ: Sine of angle θ = opposite / hypotenuse.
cos θ: Cosine of angle θ = adjacent / hypotenuse.
tan θ: Tangent of angle θ = opposite / adjacent.
Inverse trigonometric function: Used to find an angle from a ratio, e.g., sin⁻¹, cos⁻¹, tan⁻¹.

Slides

What is Trigonometry?

Trigonometry is the mathematics of angles in triangles.
It relates side lengths and angles in right-angled triangles.
The three main functions are sine (sin), cosine (cos), and tangent (tan).

What does the word 'trigonometry' come from?

Greek words meaning 'triangle' and 'measure'
Latin words meaning 'angle' and 'side'
Egyptian words meaning 'pyramid' and 'ratio'
Arabic words meaning 'right angle' and 'length'

Labelling Sides of a Right-Angled Triangle

The hypotenuse (H) is the longest side, opposite the right angle.
The opposite side (O) is opposite the given angle θ.
The adjacent side (A) is next to the given angle θ.

In a right-angled triangle, the side opposite the right angle is called $the ____$ .

SOHCAHTOA – The Ratios

Sin θ = Opposite / Hypotenuse (SOH)
Cos θ = Adjacent / Hypotenuse (CAH)
Tan θ = Opposite / Adjacent (TOA)

Which of these are correct?

Sin θ = Opposite / Hypotenuse
Cos θ = Opposite / Adjacent
Tan θ = Adjacent / Hypotenuse
Tan θ = Opposite / Adjacent

Finding a Missing Length – Step by Step

Step 1: Label sides H, O, A relative to the given angle.
Step 2: Use SOHCAHTOA to choose sin, cos or tan.
Step 3: Write the formula and substitute known values.
Step 4: Rearrange to solve for the unknown side.
Step 5: Use calculator and round to 3 significant figures.

If you know the adjacent side and want to find the hypotenuse, which ratio do you use?

sin
cos
tan
Pythagoras

Worked Example – Finding a Length

Triangle: angle 43°, adjacent $= 9$ cm, find opposite (x).
We have A and want O → use tan (TOA).
$\tan (43^{\circ}) = x / 9$ → $x = 9 \times \tan (43^{\circ})$ .
Calculator gives $x = 8.3926$ ... → round to 3 s.f. $= 8.39$ cm.

In the example, the value of x $is ____ cm (3 s.f.)$ .

Finding a Missing Angle – Step by Step

Step 1: Label sides H, O, A.
Step 2: Use SOHCAHTOA to choose sin, cos or tan.
Step 3: Write the ratio with the unknown angle.
Step 4: Use inverse function (sin⁻¹, cos⁻¹, tan⁻¹) on calculator.
Step 5: Round answer to 1 decimal place.

To find an angle when you know opposite and adjacent, which inverse function do you use?

sin⁻¹
cos⁻¹
tan⁻¹
any of them

Worked Example – Finding an Angle

Triangle: adjacent $= 8$ cm, hypotenuse $= 23$ cm, find angle y.
We have A and H → use cos (CAH).
$\cos (y) = 8 / 23$ → $y =$ cos⁻¹(8/23).
Calculator gives $y = 69.6455$ ... → round to 1 d.p. $= 69.6^{\circ}$ .

In the example, angle $y = ____$ degrees (1 d.p.).

Important Tips

SOHCAHTOA only works for right-angled triangles.
Ensure calculator is in degree mode (D or Deg).
Round lengths to 3 s.f. and angles to 1 d.p. unless told otherwise.

Which of these are true?

Trigonometry can be used in any triangle.
SOHCAHTOA is only for right-angled triangles.
Always round lengths to 3 significant figures.
Angles should be rounded to 1 decimal place.

Flashcards

What does SOHCAHTOA stand for?

$Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent$ .

What are the three trigonometric ratios for a right-angled triangle?

sin θ $= opposite/hypotenuse, \cos$ θ $= adjacent/hypotenuse, \tan$ θ $= opposite/adjacent$ .

How do you label the sides of a right-angled triangle for trigonometry?

Hypotenuse (H) is opposite the right angle; Opposite (O) is opposite the given angle; Adjacent (A) is next to the given angle.

Which trigonometric ratio do you use if you know the adjacent and hypotenuse?

Cosine (cos).

Which trigonometric ratio do you use if you know the opposite and adjacent?

Tangent (tan).

Which trigonometric ratio do you use if you know the opposite and hypotenuse?

Sine (sin).

How do you find a missing side length using trigonometry?

Label sides, choose SOHCAHTOA, substitute values, then rearrange and solve using a calculator.

If $\tan (43^{\circ}) = \frac{x}{9}$ , how do you find x?

Multiply both sides by $9: x = 9 \times \tan (43^{\circ})$ .

If $\cos (40^{\circ}) = \frac{3}{H}$ , how do you find H?

$H = 3 / \cos (40^{\circ})$ .

How do you find a missing angle using trigonometry?

Label sides, choose SOHCAHTOA, write ratio, then use inverse trig function (e.g., sin⁻¹, cos⁻¹, tan⁻¹).

If $\cos (y) = \frac{8}{23}$ , how do you find angle y?

$y =$ cos⁻¹(8/23).

To what accuracy should you round a missing side length if not specified?

Round to 3 significant figures.

To what accuracy should you round a missing angle if not specified?

Round to 1 decimal place.

What is the angle between a tangent and a radius at the point of contact?

90° (the radius is perpendicular to the tangent).

What is the angle in a semicircle?

The angle at the circumference subtended by a diameter is 90°.

In triangle XYZ, Y is due west of Z and Z is due south of X. What is angle YXZ?

Angle YXZ is found using tan⁻¹(YZ/XZ) or similar, depending on given sides.

How do you find the bearing of Y from X if $XY = 39$ km and $XZ = 21$ km?

Use trigonometry to find angle YXZ, then bearing $= 180^{\circ} +$ angle YXZ (or adjust based on quadrant).

In a right-angled triangle, if you know the opposite and adjacent, which inverse function do you use?

tan⁻¹ (inverse tangent).

In a right-angled triangle, if you know the adjacent and hypotenuse, which inverse function do you use?

cos⁻¹ (inverse cosine).

In a right-angled triangle, if you know the opposite and hypotenuse, which inverse function do you use?

sin⁻¹ (inverse sine).

What is the first step when using trigonometry to find a missing side or angle?

Label the sides of the triangle as H, O, and A relative to the given angle.

What does 'Deg' on a calculator indicate?

The calculator is set to measure angles in degrees.

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