Mathematics · Lesson 20

Trigonometry: ratios, sine & cosine rules, graphs

JAMBWAECNECO

What the syllabus asks

JAMB: Trigonometry: trigonometric ratios of angles, angles of elevation and depression, bearings, areas and solutions of triangles, graphs of sine and cosine, sine and cosine formulae

WAEC: Trigonometry: sine, cosine and tangent of acute angles, angles of 30°, 45°, 60°, trigonometric ratios of angles 0° to 360°, sine and cosine rules, area of a triangle (½ab sin C), graphs of sine and cosine, solving simple trigonometric equations

NECO: Trigonometry: sine, cosine and tangent of acute angles, angles of 30°, 45°, 60°, trigonometric ratios of angles 0° to 360°, sine and cosine rules, area of a triangle (½ab sin C), graphs of sine and cosine, solving simple trigonometric equations

What you will learn

Define sine, cosine and tangent and use them to find sides and angles in right-angled triangles.
Recall and use exact values of the ratios of 30°, 45° and 60°.
Find the ratios of angles from 0° to 360° using quadrants and the CAST rule.
Use sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ to find one ratio from another.
Apply the sine rule, including the ambiguous case, and the cosine rule.
Calculate the area of a triangle using ½ab sin C.
Draw and read graphs of y = sin x, y = cos x and combinations, and solve simple trigonometric equations.

Key points

Label O, A, H relative to the angle you are using; H is opposite the right angle.
sin = O/H, cos = A/H, tan = O/A (SOH CAH TOA).
Use sin⁻¹, cos⁻¹, tan⁻¹ to find angles; keep the calculator in degree mode.
Exact values: sin 30° = cos 60° = ½; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = 1/√2; tan 45° = 1.
sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ.
CAST: All, Sine, Tan, Cos positive in quadrants 1 to 4; use the reference angle.
Sine rule for a side–angle pair; watch for the ambiguous case with SSA.
Cosine rule for SAS and SSS; cos of an obtuse angle is negative.
Area = ½ab sin C with C between a and b.
sin x and cos x have amplitude 1 and period 360°; solve equations with reference angle plus CAST.

Start here: Naming the sides of a right-angled triangle

Trigonometry means “measuring triangles”. Surveyors use it to map land in Abuja, engineers use it to design the roof of a house, and pilots use it to plan a landing at Murtala Muhammed Airport. All of it starts with one simple idea: in a right-angled triangle, the shape is fixed by its angles, so the ratios of its sides depend only on the angles.

Before we can use those ratios, we must name the sides correctly. Pick one of the two acute angles and call it θ (theta). Then:

The hypotenuse (H) is the longest side, always opposite the right angle. It never changes.
The opposite side (O) is the side facing θ, across the triangle from it.
The adjacent side (A) is the side next to θ that is not the hypotenuse.

Here is the important point: the opposite and adjacent sides swap if you move to the other acute angle. The side opposite θ is adjacent to the other angle. So always mark the angle you are working with first, and then label the three sides.

Think of a ladder leaning against a wall of a bungalow in Enugu. The ladder is the hypotenuse. The height it reaches up the wall is opposite the angle the ladder makes with the ground. The distance from the foot of the ladder to the wall is adjacent to that angle.

Worked example 20.1

In triangle PQR, angle Q = 90°, PQ = 5 cm, QR = 12 cm and PR = 13 cm. Name the opposite, adjacent and hypotenuse for angle P.

The hypotenuse is opposite the right angle at Q, so H = PR = 13 cm.
The side facing P is QR, so O = 12 cm.
The remaining side touching P is PQ, so A = 5 cm.

O = QR = 12, A = PQ = 5, H = PR = 13

⚠️ Common mistake: calling the hypotenuse “adjacent” because it also touches the angle. The hypotenuse is always the side opposite the right angle and gets its own name.

Try it yourself. In the same triangle PQR, name O, A and H for angle R.

Show solution

H = PR = 13; the side facing R is PQ, so O = 5; the side next to R is QR, so A = 12.

Continue this lesson in the free app →

Practice questions

1. A ladder 8 m long makes 70° with the ground. How high up the wall does it reach?

Show worked answer

8 sin 70° = 8 × 0.9397 = 7.52 m.

2. If sin x = 12/13 and x is acute, find cos x and tan x.

Show worked answer

cos x = 5/13, tan x = 12/5.

3. Evaluate exactly: cos 30° sin 60° − sin 30° cos 60°.

Show worked answer

3/4 − 1/4 = 1/2.

4. Find cos 240° and sin 330° exactly.

Show worked answer

−1/2 and −1/2.

5. In triangle ABC, A = 35°, B = 80°, a = 10 cm. Find b.

Show worked answer

b = 10 sin 80°/sin 35° = 9.848/0.5736 = 17.2 cm.

Key words

Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Opposite side: The side facing the angle being considered.
Adjacent side: The side next to the angle that is not the hypotenuse.
Sine (sin): opposite ÷ hypotenuse for an acute angle in a right-angled triangle.
Cosine (cos): adjacent ÷ hypotenuse.
Tangent (tan): opposite ÷ adjacent; also equals sin θ ÷ cos θ.
Quadrant: One of the four 90° regions into which the axes divide the plane.
CAST rule: Memory aid for which ratios are positive in each quadrant.
Reference (basic) angle: The acute angle a rotating arm makes with the x-axis.
Sine rule: a/sin A = b/sin B = c/sin C for any triangle.
Cosine rule: a² = b² + c² − 2bc cos A for any triangle.
Included angle: The angle between two given sides.

Learn Trigonometry: ratios, sine & cosine rules, graphs step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android