phimaths Start free

Engineering and Science Admissions Test

ESAT Maths 1 Trigonometry questions

5 original ESAT Maths 1 trigonometry questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: Exact values of sin, cos and tan for 30°, 45°, 60°Combine the powers, then use the range of sin or cos

Question 1ESAT Maths 1Warm-up

What is the exact value of 2cos⁡230∘−cos⁡30∘tan⁡60∘2\cos^2 30^\circ - \cos 30^\circ \tan 60^\circ?
  1. A−2-2
  2. B−1-1
  3. C00
  4. D12\dfrac{1}{2}
  5. E11
  6. F22
  7. G33
Show the answer and worked solution

Answer: C · Method: Exact values of sin, cos and tan for 30°, 45°, 60°

  1. cos⁡230∘=34\cos^2 30^\circ = \dfrac{3}{4} and cos⁡30∘tan⁡60∘=32\cos 30^\circ \tan 60^\circ = \dfrac{3}{2}.
  2. So the value is 2×34−1×32=02 \times \dfrac{3}{4} - 1 \times \dfrac{3}{2} = 0.

Why the tempting wrong answers are wrong

  • B: Check your exact values: sin⁡30∘=12\sin 30^\circ = \tfrac{1}{2} and sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt{3}}{2}, not the other way round.

Try it interactively →

Question 2ESAT Maths 1Warm-up

What is the exact value of 2cos⁡230∘+sin⁡30∘cos⁡60∘2\cos^2 30^\circ + \sin 30^\circ \cos 60^\circ?
  1. A34\dfrac{3}{4}
  2. B54\dfrac{5}{4}
  3. C74\dfrac{7}{4}
  4. D94\dfrac{9}{4}
  5. E114\dfrac{11}{4}
Show the answer and worked solution

Answer: C · Method: Exact values of sin, cos and tan for 30°, 45°, 60°

  1. cos⁡230∘=34\cos^2 30^\circ = \dfrac{3}{4} and sin⁡30∘cos⁡60∘=14\sin 30^\circ \cos 60^\circ = \dfrac{1}{4}.
  2. So the value is 2×34+1×14=742 \times \dfrac{3}{4} + 1 \times \dfrac{1}{4} = \dfrac{7}{4}.

Why the tempting wrong answers are wrong

  • A: Check your exact values: sin⁡30∘=12\sin 30^\circ = \tfrac{1}{2} and sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt{3}}{2}, not the other way round.

Try it interactively →

Question 3ESAT Maths 1Warm-up

What is the exact value of 4cos⁡260∘−2sin⁡60∘cos⁡30∘4\cos^2 60^\circ - 2\sin 60^\circ \cos 30^\circ?
  1. A−52-\dfrac{5}{2}
  2. B−32-\dfrac{3}{2}
  3. C−12-\dfrac{1}{2}
  4. D00
  5. E12\dfrac{1}{2}
  6. F32\dfrac{3}{2}
  7. G52\dfrac{5}{2}
  8. H72\dfrac{7}{2}
Show the answer and worked solution

Answer: C · Method: Exact values of sin, cos and tan for 30°, 45°, 60°

  1. cos⁡260∘=14\cos^2 60^\circ = \dfrac{1}{4} and sin⁡60∘cos⁡30∘=34\sin 60^\circ \cos 30^\circ = \dfrac{3}{4}.
  2. So the value is 4×14−2×34=−124 \times \dfrac{1}{4} - 2 \times \dfrac{3}{4} = -\dfrac{1}{2}.

Why the tempting wrong answers are wrong

  • F: Check your exact values: sin⁡30∘=12\sin 30^\circ = \tfrac{1}{2} and sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt{3}}{2}, not the other way round.

Try it interactively →

Question 4ESAT Maths 1Standard

What is the minimum value of 3sin⁡x×4−sin⁡x3^{\sin x} \times 4^{-\sin x} for 0∘≤x≤360∘0^\circ \le x \le 360^\circ?
  1. A34\dfrac{3}{4}
  2. B11
  3. C43\dfrac{4}{3}
  4. D74\dfrac{7}{4}
  5. E114\dfrac{11}{4}
  6. F154\dfrac{15}{4}
  7. G194\dfrac{19}{4}
  8. H1212
Show the answer and worked solution

Answer: A · Method: Combine the powers, then use the range of sin or cos

  1. 3sin⁡x×4−sin⁡x=(34)sin⁡x3^{\sin x} \times 4^{-\sin x} = \left(\tfrac{3}{4}\right)^{\sin x}, and −1≤sin⁡x≤1-1 \le \sin x \le 1.
  2. Since 34<1\tfrac{3}{4} < 1, the minimum comes at sin⁡x=1\sin x = 1: 34\dfrac{3}{4}.

Why the tempting wrong answers are wrong

  • B: That's the value when sin⁡x=0\sin x = 0.
  • C: Combine first: it's (34)sin⁡x\left(\tfrac{3}{4}\right)^{\sin x}, and sin⁡x\sin x can be −1-1 as well as 11.

Try it interactively →

Question 5ESAT Maths 1Standard

What is the maximum value of 4sin⁡x×6−sin⁡x4^{\sin x} \times 6^{-\sin x} for 0∘≤x≤360∘0^\circ \le x \le 360^\circ?
  1. A23\dfrac{2}{3}
  2. B11
  3. C32\dfrac{3}{2}
  4. D22
  5. E52\dfrac{5}{2}
  6. F2424
Show the answer and worked solution

Answer: C · Method: Combine the powers, then use the range of sin or cos

  1. 4sin⁡x×6−sin⁡x=(46)sin⁡x4^{\sin x} \times 6^{-\sin x} = \left(\tfrac{4}{6}\right)^{\sin x}, and −1≤sin⁡x≤1-1 \le \sin x \le 1.
  2. Since 46<1\tfrac{4}{6} < 1, the maximum comes at sin⁡x=−1\sin x = -1: 32\dfrac{3}{2}.

Why the tempting wrong answers are wrong

  • A: Combine first: it's (46)sin⁡x\left(\tfrac{4}{6}\right)^{\sin x}, and sin⁡x\sin x can be −1-1 as well as 11.
  • B: That's the value when sin⁡x=0\sin x = 0.

Try it interactively →

There are 39 trigonometry questions on phimaths.

Free, no sign-up. Get one wrong and you get a new question that needs the same method.

Start practising →

More ESAT Maths 1 topics