Trigonometry – GCSE Maths

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Introduction

  • Trigonometry is all about Triangles.
  • It is a branch of mathematics that deals with the relationships between the angles and sides of triangles—especially right-angled triangles.

Image of different triangles such as isosceles or right angled triangles showing angles nad sides missing in order to explain the uses of trigonometry

Basics of Trigonometry

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

1. Why do we use it?

To find:

  • How long a side is
  • What an angle is

—when we have the values of some other parts of the triangle.

2. The Three main Functions:

In a right-angled triangle:

  • sin (as: “sine”)
  • cos (as: “cosine”)
  • tan (as: “tangent”)

They are simply the ratios (fractions) of the given triangle’s sides.

3. All about Triangles:

Triangles are three-sided polygons with several important properties. Here are some key properties of triangles:-

Basic Properties-

  • A triangle has three sides, three vertices, and three angles.
  • The sum of the interior angles is always 180°.
  • The sum of the exterior angles is always 360°.

Side Length Rule (Triangle Inequality Theorem)Image showing that sum of 2 sides of a triangle is always greater than the third side4. Types of Triangles:

Angles Of Elevation & Depression

Definitions-

  • Angle of Elevation: The angle formed between the horizontal line (eye level) and the line of sight when an observer looks upwards at an object.
  • Angle of Depression: The angle formed between the horizontal line (eye level) and the line of sight when an observer looks downwards at an object.

Image showing angles of elevation and angles of depression with respect to a human and a puppy and an aircraft to understand Trigonometry for gcse maths

Key Points-

  • Both angles are measured from the horizontal (eye level).
  • They are always between 0° and 90°.
  • The angle of elevation and depression are congruent (equal) when the observer and object are at the same horizontal level (i.e., in symmetric positions).

Real Life Applications-

  • Angle of Elevation: Used in measuring heights of buildings, mountains, or trees.
  • Angle of Depression: Used in aviation (pilots landing planes), navigation, or determining distances between objects at different heights.

Step by Step Procedure-

  • Step#1: Draw a Diagram
  • Step#2: Identify known and unknown values
  • Step#3: Choose the Right Trigonometric Ratio
  • Step#4: Solve for the Unknown
  • Step#5: Check for Angle of Depression

certified Physics and Maths tutorSolved Example:

Example: “A bird sits on a tree 10m high. A man 20 m away looks up at the bird.”

Solution:

Step#1: Draw a Diagram-

  • Sketch the scenario based on the problem statement.
  • Label-
    • The observer’s eye level (horizontal line).
    • The line of sight (angle of elevation or depression).
    • The height (vertical side) and distance (horizontal side).

Image showing angle of elevation for solved example for explaining concept with step by step solved example

Step#2: Identify Known & Unknown Values-

  • Given:
    • Distance from observer to object (adjacent side).
    • Height (opposite side).
    • Angle (if given).
  • Find:
    • The missing side or angle.

Example:

  • Given:Solution for step by step solved example of angle of elevation concept for trigonometry
  • Find: Angle of Elevation (θ).

Step#3: Choose the Right Trigonometric Ratio-

  • SOH-CAH-TOA helps decide which ratio to use:
    • Sine (sinθ) = Opposite / Hypotenuse
    • Cosine (cosθ) = Adjacent / Hypotenuse
    • Tangent (tanθ) = Opposite / Adjacent

In our example:

  • We have opposite (height) = 10m and adjacent (distance) = 20m.
  • Use tangent-

Solution for step by step solved example of angle of elevation concept for trigonometry

Step#4: Solve for the Unknown-

  • If finding an angle, use inverse trig functions (tan⁻¹, sin⁻¹, cos⁻¹).
  • If finding a side, rearrange the formula.

Example (continued):

  • To find θ:
  • θ = tan−1(0.5) ≈ 26.57°

Step#5: Check for Angle of Depression-

  • If the problem involves looking downward, the steps are the same, but the angle is measured below the horizontal.

Key Fact:

  • Angle of elevation from point A to B = Angle of depression from B to A (they are equal due to alternate angles).

Therefore,

Angle of Elevation = Angle of Depression

Hence,

Angle of depression ≈ 26.57°

certified Physics and Maths tutorSolved Example:

Example: A bird is perched on a 15-meter-high tree. It spots a worm on the ground 9 meters away from the base of the tree. What is the angle of depression from the bird to the worm?

Solution:

Step#1: Draw a Diagram-

  • Sketch the scenario based on the problem statement.
  • Label:
    • The observer’s eye level (horizontal line).
    • The line of sight (angle of elevation or depression).
    • The height (vertical side) and distance (horizontal side).Image showing 2 sides of an imaginary triangle for step by step solved example of angle of elevation for trigonometry

Step#2: Identify Known & Unknown Values-

  • Given:
    • Distance from observer to object (adjacent side).
    • Height (opposite side).
    • Angle (if given).
  • Find:
    • The missing side or angle.

Example:

  • Given:
    • Distance (adjacent) = 9m
    • Height (opposite) = 15m
  • Find: Angle of depression(θ).

Step#3: Choose the Right Trigonometric Ratio-

  • SOH-CAH-TOA helps decide which ratio to use:
    • Sine (sinθ) = Opposite / Hypotenuse
    • Cosine (cosθ) = Adjacent / Hypotenuse
    • Tangent (tanθ) = Opposite / Adjacent

In our example:

  • We have opposite (height) = 15m and adjacent (distance) = 9m.
  • Use tangent-Solution for step by step solved example of angle of elevation concept for trigonometry

Step#4: Solve for the Unknown-

  • If finding an angle, use inverse trig functions (tan⁻¹, sin⁻¹, cos⁻¹).
  • If finding a side, rearrange the formula.

Example (continued):

  • To find θ:
  • θ = tan−1(1.67) ≈ 59.3°

Step#5: Check for Angle of Elevation-

  • If the problem involves looking downward, the steps are the same, but the angle is measured below the horizontal.

Key Fact:

  • Angle of elevation from point A to B = Angle of depression from B to A (they are equal due to alternate angles).

Therefore,

Angle of Elevation = Angle of Depression

Hence,

Angle of depression ≈ 59.3°

Triangles Exact Values

Image of triangle showing hypotenuse, adjacent and opposite sides

Let us understand about some important ratios in brief:image showing relation of sine, cos and tan with hypotenuse, adjacent and opposite sides of a triangleWhere the terms are denoted as:

  • Opposite = side opposite the angle
  • Adjacent = side next to the angle (not the hypotenuse)
  • Hypotenuse = the longest side (opposite the 90° angle

Image of Table of angles for triangle exact values with sine, cos and tan with sides of triangles

  • Tip: We have to summarize this table given above to solve each of the question accurately.

certified Physics and Maths tutorSolved Example:

Example: In a right triangle, the angle is 30° and the adjacent side is 6 units. Find the opposite side.

Solution:Solution step by step solved example for trigonometrySo, therefore we got an answer to our question that is: Solution step by step solved example

certified Physics and Maths tutorSolved Example:

Example: In a right triangle, the angle is 30° and the opposite side is 9 units. Find the opposite side.

Solution:

Given:

  • Angle = 30°
  • Adjacent side = 6 units

We know that,Solution step by step solved example for trigonometry

So, therefore we got an answer to our question that is:

Solution step by step solved example for trigonometry

certified Physics and Maths tutorSolved Example:

Problem: A shed roof makes an angle of 41° with the horizontal. Given that the width of the shed is 6 m and the length of its slope is 4 m. Calculate the height of the roof.

Solution:

Given:

  • Angle (θ) = 41° (between the roof and the horizontal)
  • Slope length (L) = 4 m (the hypotenuse of the right triangle formed by the roof)
  • Width (W) = 6 m (total horizontal span of the shed)

The width of the shed (6 m) is the total span, but the roof slope only covers half of this (since it’s a symmetrical shed roof). 

Final solved step by step examples of trigonometry for gcse maths

The height of the roof is approximately 2.624 meters.

certified Physics and Maths tutorSolved Example:

Problem: A zip wire runs between two poles 45m apart. The zip wire is at an angle of 10° to the horizontal. Calculate the length of the zip wire.

image for Final solved step by step examples of trigonometry for gcse maths

Solution:

Given:

  • Angle (θ) = 10° (between the zip wire and the length)
  • Width (W) = 25 m (Distance between two poles)

The width of the shed (6 m) is the total span, but the roof slope only covers half of this (since it’s a symmetrical shed roof).Answer image for Final solved step by step examples of trigonometry for gcse mathsThe length of the zip wire is approximately 25.38 meters.

certified Physics and Maths tutorSolved Example:

Problem: Triangle ABC is an isosceles. Calculate the height of the given triangle.image for Final solved step by step examples of trigonometry for gcse mathsSolution:

Given:

  • Angle (θ) = 71° (between the two sides)
  • Side length = 12 cm (Distance between two poles)

The width of the shed (6 m) is the total span, but the roof slope only covers half of this (since it’s a symmetrical shed roof).Answer image for Final solved step by step examples of gcse mathsThe height of the triangle ABC is approximately 17.4 centimeters.

 

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