Trigonometry 2 (Edexcel GCSE Maths): Revision Notes
Trigonometry 2
Finding missing sides using trigonometric ratios
You can use trigonometric ratios to calculate the length of unknown sides in right-angled triangles. This method works when you know the length of one side and the size of one of the acute angles.
The key is to correctly identify which sides of the triangle are the opposite, adjacent, and hypotenuse relative to the given angle, then choose the appropriate trigonometric ratio.
Before using any trigonometric ratio, you must first correctly label the sides of the triangle relative to the angle you're working with. This is the most crucial step that determines whether you'll get the right answer.
Labelling triangle sides
Before using any trigonometric ratio, you must label the sides of the triangle relative to the angle you're working with:
- Hypotenuse: The longest side, always opposite the right angle
- Opposite: The side directly across from your given angle
- Adjacent: The side next to your given angle (but not the hypotenuse)
The labels "opposite" and "adjacent" change depending on which angle you're working with. Always relabel the triangle if you switch to a different angle in your calculations.
Choosing the correct ratio
Once you've labelled the sides, select the trigonometric ratio that connects the sides you know and need to find:
Worked example walkthrough
When calculating a missing side length, follow these systematic steps to ensure accuracy and avoid common mistakes.
Worked Example: Finding the Opposite Side
Given: Right-angled triangle with a 40° angle and hypotenuse of 10cm Find: The opposite side
Step 1: Label the triangle sides relative to the 40° angle
- Hypotenuse = 10cm (given)
- Opposite = ? (what we're finding)
- Adjacent = not needed for this problem
Step 2: Write down the appropriate trigonometric ratio We have hypotenuse and need opposite, so use:
Step 3: Substitute the known values
Step 4: Rearrange to solve for the unknown side
Step 5: Calculate using your calculator (to 3 s.f.)
Calculator Accuracy Tip
Always write down at least four figures from your calculator display before rounding your final answer to the required number of significant figures. This ensures accuracy in your calculations and prevents rounding errors from accumulating.
Angles of elevation and depression
Angles of elevation and depression appear in real-world trigonometry problems and represent angles measured from the horizontal.
Key definitions
- Angle of elevation: The angle measured upwards from the horizontal to an object above eye level
- Angle of depression: The angle measured downwards from the horizontal to an object below eye level
Critical Principles for Elevation and Depression Problems
- Both types of angles are always measured from the horizontal line, not from vertical
- When dealing with elevation and depression problems, angles on opposite sides of parallel horizontal lines are alternate angles, meaning they are equal
- These problems often involve scenarios like ships, buildings, aircraft, or other real-world situations
Solving elevation and depression problems
Step-by-Step Approach:
- Draw a clear diagram showing the horizontal reference line
- Mark the angles of elevation or depression correctly
- Identify alternate angles where appropriate
- Label the triangle sides relative to your chosen angle
- Apply the appropriate trigonometric ratio to find the missing measurement
Practice problems approach
Key Strategies for Triangle Problems:
When working through triangle problems, follow this systematic approach:
- Always start by identifying which side you need to find
- Check whether you're given an angle and one side length
- Be particularly careful when the unknown side is the hypotenuse - it will appear in the denominator when you substitute values
- Remember to give answers to the specified degree of accuracy (usually 1 decimal place or 3 significant figures)
Key Points to Remember:
- Label triangle sides relative to your given angle before choosing a trigonometric ratio
- SOH CAH TOA helps you remember which ratio to use: , ,
- Angles of elevation and depression are always measured from the horizontal, not the vertical
- Write down extra figures from your calculator before rounding to avoid accuracy errors
- Check your answer makes sense - shorter sides should be smaller than the hypotenuse