Leaving Certificate • Ordinary Level

Trigonometry

Exam-focused notes, key formulas, common mistakes, and interactive practice cards (with solutions).

Overview Formulas Question Types Common Mistakes Practice Checklist

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Overview: what OL trigonometry usually tests Notes

• Solving right-angled triangles using \(\sin\), \(\cos\), \(\tan\).
• Using Pythagoras’ theorem in right-angled triangle contexts.
• Angles of elevation and depression in real-life diagrams.
• Solving non-right-angled triangles using the sine rule and cosine rule.
• Area of a triangle using \(\tfrac{1}{2}ab\sin C\) in multi-step problems.

Calculator: Set your calculator to DEG (degrees). Rounding instructions matter.

Key formulas (must-know) Formulas

Right-angled triangles (SOHCAHTOA)

\[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \]

Pythagoras’ theorem

\[ a^2+b^2=c^2 \]

Cosine rule (non-right-angled triangles)

\[ a^2=b^2+c^2-2bc\cos A \]

Sine rule (non-right-angled triangles)

\[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \]

Area of a triangle using sine

\[ \text{Area}=\tfrac12 ab\sin C \]

Question types (what to recognise quickly) Exam Skills

1) Right-angled triangle (find a side)

• Given: one angle + one side
• Find: another side

Method: label opposite/adjacent/hypotenuse → choose ratio → solve.

2) Right-angled triangle (find an angle)

• Given: two sides
• Find: an angle

Method: form a ratio → use \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).

3) Elevation / depression

• Often a “line of sight” diagram
• Height and distance from a horizontal

Tip: draw a horizontal line; use alternate angles where needed.

4) Pythagoras then trig

• Part (a): find a missing side using \(a^2+b^2=c^2\)
• Part (b): use trig ratios with the result

Tip: keep intermediate answers to 3 significant figures (unless told otherwise).

5) Cosine rule

• Non-right-angled triangle
• Two sides and the included angle known

Tip: watch brackets: \(b^2+c^2-2bc\cos A\).

6) Sine rule

• Non-right-angled triangle
• At least one opposite side–angle pair known

Tip: always match a side with its opposite angle.

7) Area using \(\tfrac12 ab\sin C\)

• Two sides and the included angle given
• Often embedded in longer questions

Tip: include correct units (e.g. \(\text{m}^2\)).

8) Multi-step “real-life” problem

• Mix of Pythagoras + trig + sine/cosine rule
• Usually high-value marks

Tip: answer parts in order; method marks are key at OL.

Common mistakes (avoid losing easy marks) Warnings

• Calculator not in DEG mode.
• Mixing up opposite and adjacent.
• Using sine/cosine rule in a right-angled triangle.
• Forgetting units for length/area.
• Ignoring rounding instructions (nearest degree / 1 d.p. / nearest metre).

Practice: exam-style questions (with reveal solutions) Interactive

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Question 1

In a right-angled triangle, the hypotenuse is \(12\text{ cm}\) and an acute angle is \(35^\circ\). Find the length of the side opposite the \(35^\circ\) angle. Give your answer correct to 1 decimal place.

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Question 2

From a point on level ground, the angle of elevation to the top of a building is \(28^\circ\). The point is \(40\text{ m}\) from the base of the building. Find the height of the building, correct to the nearest metre.

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Question 3

In triangle \(ABC\), \(AB=9\text{ cm}\), \(AC=7\text{ cm}\), and \(\angle A=52^\circ\). Find \(BC\), correct to 1 decimal place.

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Question 4

Two sides of a triangle are \(a=11\text{ m}\) and \(b=8\text{ m}\) and the included angle is \(39^\circ\). Find the area of the triangle, correct to the nearest square metre.

Student checklist (ready for revision) Self-check

I can label opposite/adjacent/hypotenuse correctly. I can choose between \(\sin\), \(\cos\), and \(\tan\) for a right-angled triangle. I can use \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) to find angles. I can use Pythagoras’ theorem confidently. I recognise elevation/depression questions and draw the correct triangle. I know when to use the cosine rule (2 sides + included angle). I know when to use the sine rule (an opposite side–angle pair). I can find area using \(\tfrac12 ab\sin C\). I round answers correctly and include units. I keep my calculator in DEG mode.

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