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1.7 Trigonometry

1.7 Trigonometry

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Question 45
a.

Prove that

sin⁡2θ1+cos⁡2θ≡tan⁡θ\displaystyle \frac{\sin 2\theta}{1 + \cos 2\theta} \equiv \tan\theta1+cos2θsin2θ​≡tanθ

A student is attempting to solve the equation

sin⁡2θ1+cos⁡2θ=2sin⁡θ\displaystyle \frac{\sin 2\theta}{1 + \cos 2\theta} = 2\sin\theta1+cos2θsin2θ​=2sinθ for 0°≤θ≤360° 0° \leq \theta \leq 360°\,0°≤θ≤360°

They use the result from part (a), and write the following incorrect solution.

Step 1: tan⁡θ=2sin⁡θ\tan\theta = 2\sin\thetatanθ=2sinθ

Step 2: sin⁡θcos⁡θ=2sin⁡θ\dfrac{\sin\theta}{\cos\theta} = 2\sin\thetacosθsinθ​=2sinθ

Step 3: 1cos⁡θ=2\dfrac{1}{\cos\theta} = 2cosθ1​=2

Step 4: cos⁡θ=12\cos\theta = \dfrac{1}{2}cosθ=21​

Step 5: θ=60°\theta = 60°θ=60°, 300°300°300°

[3]
b(i).

Explain the error the student has made between Step 2 and Step 3.

[2]
b(ii).

State the complete set of solutions of the equation for 0°≤θ≤360°0° \leq \theta \leq 360°0°≤θ≤360°.

[2]
Markscheme

1.7 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.7 Trigonometry

317 exam-style questions on OCR (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.

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