Further Mathematics · Lesson 10

Trigonometry: identities, compound angles, equations

WAECNECO

What the syllabus asks

WAEC: Pure Mathematics: trigonometry — identities, compound and multiple angles, sin(A ± B), cos(A ± B), tan(A ± B), double angles, a cos θ + b sin θ, solution of trigonometric equations for 0° ≤ θ ≤ 360°

NECO: Pure Mathematics: trigonometry — identities, compound and multiple angles, sin(A ± B), cos(A ± B), tan(A ± B), double angles, a cos θ + b sin θ, solution of trigonometric equations for 0° ≤ θ ≤ 360°

What you will learn

Use special-angle values and the ASTC rule.
State and use the Pythagorean identities.
Prove trigonometric identities by a clear, one-sided method.
Use compound-angle and double-angle formulae, including exact values.
Express a cosθ + b sinθ as R cos(θ − α) and find maxima.
Solve trigonometric equations in a given range, including quadratic types.

Key points

ASTC: All (1st), Sin (2nd), Tan (3rd), Cos (4th) are positive.
sin² + cos² = 1; 1 + tan² = sec²; 1 + cot² = cosec².
Prove identities on one side only; convert to sin and cos.
sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB.
tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).
sin2A = 2sinA cosA; cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A.
a cosθ + b sinθ = R cos(θ − α), R = √(a² + b²), tanα = b/a; max R.
Equations: basic angle, ASTC, all solutions in range; factorise, never divide; reject |sin|,|cos| > 1.

Start here: Special angles and the four quadrants

Trigonometry in Further Maths extends the ratios beyond 90°. Imagine a point P moving round a circle of radius 1 centred at the origin. If OP makes angle θ with the positive x-axis, then the coordinates of P are (cos θ, sin θ). This definition works for every angle, not just acute ones.

From this picture, the signs follow: in the first quadrant both coordinates are positive, so all ratios are positive. In the second, x < 0 and y > 0, so only sin is positive. In the third, both are negative, so only tan (y/x) is positive. In the fourth, only cos. The memory aid is ASTC ("All Students Take Chemistry"), going anticlockwise from the first quadrant.

θ	0°	30°	45°	60°	90°
sin	0	1/2	√2/2	√3/2	1
cos	1	√3/2	√2/2	1/2	0
tan	0	√3/3	1	√3	undefined

For any angle, find the basic angle (the acute angle to the x-axis), take its ratio, and attach the sign from ASTC.
2nd quadrant: θ = 180° − basic · 3rd: 180° + basic · 4th: 360° − basic

Worked example 10.1

Find the exact value of cos 210°.

210° is in the 3rd quadrant: basic angle 210° − 180° = 30°.
cos is negative in the 3rd quadrant.
cos 210° = −cos 30° = −√3/2.

−√3/2

Worked example 10.2

Find tan 315°.

4th quadrant, basic angle 45°. tan is negative there.
tan 315° = −1.

−1

⚠️ Common mistake: using 210° − 90° = 120° as the basic angle. The basic angle is always measured to the x-axis (180° or 360°).

Try it yourself. Find exact values of sin 120° and cos 300°.

Show solution

sin 120° = √3/2; cos 300° = 1/2.

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Practice questions

1. Find the exact value of cos 135°.

Show worked answer

−√2/2.

2. Prove that (sin θ + cos θ)² = 1 + sin 2θ.

Show worked answer

sin² + 2sin cos + cos² = 1 + sin 2θ.

3. Find sin 105° exactly.

Show worked answer

sin(60° + 45°) = (√6 + √2)/4.

4. If cos A = 12/13 (A acute), find sin 2A.

Show worked answer

sin A = 5/13; sin 2A = 120/169.

5. Solve 2 sin θ − √2 = 0 for 0° ≤ θ ≤ 360°.

Show worked answer

sin θ = √2/2: 45°, 135°.

Key words

Identity: An equation true for every value of the variable for which both sides are defined.
Equation: A statement true only for particular values, which we solve for.
sec θ: 1/cos θ.
cosec θ: 1/sin θ.
cot θ: 1/tan θ = cos θ/sin θ.
Compound angle: An angle formed by adding or subtracting angles, e.g. A + B.
Double angle: An angle of the form 2A.
Basic (reference) angle: The acute angle between the terminal side and the x-axis.
Principal value: The value given by the calculator’s inverse function.
Quadrant: One of the four 90° regions of the plane.
Amplitude: The greatest value of R cos(θ − α), namely R.
Exact value: A value given with surds, not a decimal.

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