Further Trigonometry questions

Questions to attempt

6 questions on Further Trigonometry, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FP1-1.1 · The t-formulae.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   Starting from $\sin\theta=2\sin\tfrac{\theta}{2}\cos\tfrac{\theta}{2}$, derive the t-formula $\sin\theta=\dfrac{2t}{1+t^2}$, where $t=\tan\tfrac{\theta}{2}$.

   (3)

   (Total for Question 1 is 3 marks)

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2. FP1-1.1 · The t-formulae.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   Given that $t=\tan\tfrac{\theta}{2}=\tfrac12$, find the exact values of $\sin\theta$ and $\cos\theta$.

   (2)

   (Total for Question 2 is 2 marks)

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3. FP1-1.2 · Applications of t-formulae to trigonometric identities.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   Prove that $\cot\tfrac{\theta}{2}-\tan\tfrac{\theta}{2}\equiv2\cot\theta$.

   (4)

   (Total for Question 3 is 4 marks)

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4. FP1-1.2 · Applications of t-formulae to trigonometric identities.

   Q4

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   4.

   Using $t=\tan\tfrac{\theta}{2}$, prove that $\operatorname{cosec}\theta+\cot\theta\equiv\cot\tfrac{\theta}{2}$.

   (3)

   (Total for Question 4 is 3 marks)

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5. FP1-1.3 · Applications of t-formulae to solve trigonometric equations.

   Q5

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   5.

   Solve $\sin x-\cos x=1$ for $0\le x<2\pi$, explaining carefully why the t-substitution alone does not give the complete solution set.

   (5)

   (Total for Question 5 is 5 marks)

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6. FP1-1.3 · Applications of t-formulae to solve trigonometric equations.

   Q6

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   6.

   Use the substitution $t=\tan\tfrac{x}{2}$ to solve $\sin x+\cos x=1$ for $0\le x<2\pi$.

   (4)

   (Total for Question 6 is 4 marks)

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FP1-1 Further Trigonometry (Further Pure 1) revision notes