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1.7 Trigonometry

1.7 Trigonometry

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Question 122

A research team is modeling the vertical displacement, HHH, of a specialized underwater sensor. The displacement is given by the function

H(θ)=sin⁡2θsec⁡θ+cos⁡2θcsc⁡θ+sin⁡θ H(\theta) = \sin 2\theta \sec \theta + \cos 2\theta \csc \theta + \sin \theta H(θ)=sin2θsecθ+cos2θcscθ+sinθ

where θ\thetaθ is the tilt angle of the sensor.

a.

Show that the expression for H(θ)H(\theta)H(θ) can be written as

H(θ)=sin⁡θ+csc⁡θ H(\theta) = \sin \theta + \csc \theta H(θ)=sinθ+cscθ

where sin⁡θ≠0\sin \theta \neq 0sinθ=0 and cos⁡θ≠0\cos \theta \neq 0cosθ=0.

[4]
bi.

A technician attempts to find the tilt angles where the displacement is exactly 4.254.254.25 units by solving the equation

sin⁡2θsec⁡θ+cos⁡2θcsc⁡θ+sin⁡θ=4.25 \sin 2\theta \sec \theta + \cos 2\theta \csc \theta + \sin \theta = 4.25 sin2θsecθ+cos2θcscθ+sinθ=4.25

for 0∘≤θ≤360∘0^{\circ} \leq \theta \leq 360^{\circ}0∘≤θ≤360∘. They produce the following solution:

Step 1: sin⁡θ+csc⁡θ=4.25\sin \theta + \csc \theta = 4.25sinθ+cscθ=4.25

Step 2: sin⁡θ+1sin⁡θ=174\sin \theta + \frac{1}{\sin \theta} = \frac{17}{4}sinθ+sinθ1​=417​

Step 3: 4sin⁡2θ−17sin⁡θ+4=04\sin^{2} \theta - 17\sin \theta + 4 = 04sin2θ−17sinθ+4=0

Step 4: sin⁡θ=4\sin \theta = 4sinθ=4 or sin⁡θ=0.25\sin \theta = 0.25sinθ=0.25

Step 5: θ=14.5∘,165.5∘\theta = 14.5^{\circ}, 165.5^{\circ}θ=14.5∘,165.5∘

Explain why the technician should reject the value sin⁡θ=4\sin \theta = 4sinθ=4 in Step 4.

[1]
bii.

Determine if there are any other reasons, based on the original expression's domain, why solutions might need to be rejected, and state the final correct solutions for the technician's equation in the range 0∘≤θ≤360∘0^{\circ} \leq \theta \leq 360^{\circ}0∘≤θ≤360∘.

[3]
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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