Trigonometry — Sin, Cos, Tan Revision Notes for Edexcel GCSE Maths

Trigonometry — Sin, Cos, Tan

Introduction to trigonometry

Trigonometry might seem like a big, intimidating word, but it's actually a powerful and useful mathematical tool. At its core, trigonometry helps us understand the relationships between the angles and sides of triangles, particularly right-angled triangles. Once you master the basics, you'll find trigonometry appearing everywhere from architecture to navigation.

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Understanding trigonometry is essential for many fields including engineering, physics, computer graphics, and even music production. The concepts you learn here will form the foundation for more advanced mathematical topics.

The three fundamental trigonometric ratios

There are three essential trigonometric functions that form the foundation of all trigonometry work. Each one creates a specific relationship between an angle in a right-angled triangle and two of its sides.

The three basic formulas are:

Sine (Sin): Links the opposite side to the hypotenuse
Cosine (Cos): Links the adjacent side to the hypotenuse
Tangent (Tan): Links the opposite side to the adjacent side

These can be expressed mathematically as:

$\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan x = \frac{\text{Opposite}}{\text{Adjacent}}$

Understanding triangle sides

Before you can use trigonometric ratios effectively, you need to understand how to identify the different sides of a right-angled triangle in relation to a specific angle.

In any right-angled triangle, there are three sides with specific names:

Hypotenuse: This is always the longest side of the triangle, positioned opposite the right angle
Opposite: This is the side that sits directly across from the angle you're working with
Adjacent: This is the side that's next to the angle you're working with (but not the hypotenuse)

chatImportant

Remember that the opposite and adjacent sides change depending on which angle you're considering, but the hypotenuse always stays the same. This is a common source of confusion for students, so always double-check your triangle labelling before proceeding with calculations.

The SOH CAH TOA mnemonic

The most popular way to remember the trigonometric ratios is through the mnemonic SOH CAH TOA. This breaks down as:

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

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Some students prefer alternative mnemonics like "Strange Orange Hamsters Creep Around Houses Tripping Over Ants" to help them remember this sequence. Find the version that works best for you and practice it until it becomes automatic.

Formula triangles method

Formula triangles provide a visual way to rearrange trigonometric equations, making it easier to solve for different variables. This method works particularly well for visual learners.

For each trigonometric ratio, you can create a triangle where:

The numerator goes on top
The denominator goes on the bottom
You cover up what you want to find

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Step-by-Step Approach Using Formula Triangles:

Label the triangle sides as O (Opposite), A (Adjacent), and H (Hypotenuse)
Write down SOH CAH TOA from memory
Decide which sides are involved in your problem
Choose the appropriate formula (SOH, CAH, or TOA)
Create the formula triangle for your chosen ratio
Cover up what you want to find with your finger
Work out the calculation using the remaining visible parts

This systematic approach ensures you don't miss any steps and helps build confidence in your problem-solving abilities.

Solving trigonometry problems

When approaching trigonometry questions, follow these systematic steps to ensure accurate results:

Identify the triangle type: Ensure you're working with a right-angled triangle
Label the sides relative to your angle of interest
Determine which ratio to use based on the sides involved
Set up your equation using the appropriate formula
Solve for the unknown using your calculator in DEG mode
Check your answer to ensure it makes sense

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For finding angles, you'll need to use the inverse functions ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ ) on your calculator. Make sure your calculator is set to degrees (DEG) mode, not radians, when working with angle measurements in degrees.

Worked examples

Let's work through some typical problems to see how these principles apply in practice.

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Worked Example 1: Finding a Missing Side

Given: A right-angled triangle with an angle of 30° and hypotenuse of 10 cm Find: The length of the opposite side

Step 1: Identify the known values

Angle = 30°
Hypotenuse = 10 cm
Need to find: Opposite side

Step 2: Choose the appropriate ratio Since we have the hypotenuse and need the opposite, we use SOH (Sine) $\sin 30° = \frac{\text{Opposite}}{10}$

Step 3: Solve for the opposite side $\text{Opposite} = 10 \times \sin 30° = 10 \times 0.5 = 5 \text{ cm}$

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Worked Example 2: Finding a Missing Angle

Given: A right-angled triangle with opposite side = 6 cm and adjacent side = 8 cm Find: The angle

Step 1: Identify the known values

Opposite = 6 cm
Adjacent = 8 cm
Need to find: Angle

Step 2: Choose the appropriate ratio Since we have opposite and adjacent, we use TOA (Tangent) $\tan \theta = \frac{6}{8} = 0.75$

Step 3: Use inverse tangent to find the angle $\theta = \tan^{-1}(0.75) = 36.87°$

Angles of elevation and depression

These concepts apply trigonometry to real-world scenarios, particularly useful in surveying, navigation, and construction.

Angle of elevation: This is the angle measured upwards from the horizontal to your line of sight. For example, when you look up at the top of a building, the angle your line of sight makes with the horizontal ground is the angle of elevation.

Angle of depression: This is the angle measured downwards from the horizontal to your line of sight. For instance, when looking down from a cliff to a boat below, the angle your line of sight makes with the horizontal is the angle of depression.

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An important relationship to remember is that the angle of elevation from point A to point B equals the angle of depression from point B to point A. This symmetry is extremely useful when solving real-world problems involving heights and distances.

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Key Points to Remember:

SOH CAH TOA is your best friend - memorise it completely as it's the key to success in trigonometry
Always identify your triangle sides relative to the angle you're working with before choosing a formula
The hypotenuse is always the longest side and sits opposite the right angle
Check your calculator is in DEG mode when working with degrees
Your answers should make sense - if a side length is negative or an angle is greater than 90°, double-check your work
Angles of elevation and depression are equal when measured between the same two points
Practice regularly - trigonometry becomes much easier with repetition and familiarity

Trigonometry — Sin, Cos, Tan (Edexcel GCSE Maths): Revision Notes