Year 9 · Trigonometry
Name the sides of a right-angled triangle
Learn how to use sine, cosine and tangent to find sides and angles in right-angled triangles, recognise surds, and work out what pay, borrowing, insurance and spending abroad really cost.
Definition
Trigonometry connects the angles of a right-angled triangle to the ratios of its sides: sine, cosine and tangent each compare a different pair of sides, so knowing one angle and one side is enough to find the rest. Alongside it you meet surds, roots such as √2 that have no exact decimal form, so writing the root keeps the value exact. The rest of the unit is the maths of money: bank accounts, pay, investments, mortgages, loans, running costs, insurance and spending abroad, where percentages and careful totals decide what things really cost.
Why Do I Have to Learn This?
Trigonometry links angles to lengths in triangles. Engineers, pilots and animators use it to work out heights and distances.
Key Rules:
- 1. In a right-angled triangle the hypotenuse faces the right angle, the opposite faces the angle you are using and the adjacent sits next to it; then sin = opposite/hypotenuse, cos = adjacent/hypotenuse and tan = opposite/adjacent.
Remember:
- •Label the triangle before you pick a ratio, and with money always separate what comes in from what goes out.
Don’t Forget...
- •Choosing sin when the two sides in the question are the opposite and the adjacent. sin links the opposite to the hypotenuse; the only ratio linking opposite and adjacent is tan, so the wrong choice gives a wrong answer even with perfect arithmetic.
- •Treating gross pay as the money you can spend. Deductions such as tax and National Insurance are taken before pay arrives, so take-home pay is gross pay minus every deduction, and budgeting from the gross figure overstates what is left.
- •Calling every root a surd, including √9. √9 is exactly 3, a whole number, so it is not a surd; a surd is a root such as √2 whose decimal never ends or repeats.
Cosine
One of the three trigonometric ratios. In a right-angled triangle, cos of an angle = adjacent ÷ hypotenuse.
cos 60° = 0.5, so when the hypotenuse is 10 cm the adjacent side is 5 cm.
Worked Example:
In a right-angled triangle, the angle at the base is 35° and the side adjacent to that angle is 8 cm. Find the length of the side opposite the 35° angle, to 1 decimal place.
Label the sides: the 8 cm side is adjacent to the 35° angle, and the side to find is opposite it.
The ratio that links opposite and adjacent is tan, so tan 35° = opposite/8.
Rearrange to make the missing side the subject: opposite = 8 × tan 35°.
tan 35° = 0.7002 to 4 decimal places, so opposite = 8 × 0.7002... = 5.6016...
Answer: 5.6 cm (to 1 decimal place)
Why It Works:
All right-angled triangles that share an angle are similar, so the ratio between any pair of their sides is a fixed number depending only on that angle: this is why sin, cos and tan work, and why one known side and one angle determine every other side. The money methods are valid for the same reason every ratio method is: a percentage is a fixed fraction of an amount and an exchange rate is a fixed ratio between currencies, so multiplying by them scales an amount exactly.
Another Example:
Amira earns £2,400 a month before deductions. Each month £310 goes in income tax, £150 in National Insurance and £120 into her pension. What is her monthly take-home pay?
Total the deductions: 310 + 150 + 120 = £580.
Subtract the deductions from her gross pay: £2,400 - £580 = £1,820.
Answer: £1,820 a month
Animated Example:
Free Name the sides of a right-angled triangle Worksheet
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Let’s Practise the Concept
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