Reciprocal and inverse trig functions Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Reciprocal and inverse trig functions questions. See exactly how to solve problems on reciprocal trig, exact values, inverse trig, identities.

reciprocal trigexact valuesinverse trigidentitiesconcepttrig equations
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find the exact value of sec⁡π3\sec \frac{\pi}{3}.

Worked solution

  1. Recall the definition of the reciprocal function

    sec⁡x=1cos⁡x\sec x=\dfrac{1}{\cos x}

    The reciprocal function is one divided by the matching basic ratio.

  2. Evaluate the underlying trigonometric ratio

    cos⁡π3=12\cos \frac{\pi}{3}=\frac{1}{2}

    Use the special triangles or the unit circle for this exact ratio.

  3. State the exact value

    sec⁡π3=2\sec \frac{\pi}{3}=2

    This is the required exact value.

Answer
22
Question 2
2 markseasy
Find the exact value of csc⁡π6\csc \frac{\pi}{6}.

Worked solution

  1. Recall the definition of the reciprocal function

    csc⁡x=1sin⁡x\csc x=\dfrac{1}{\sin x}

    The reciprocal function is one divided by the matching basic ratio.

  2. Evaluate the underlying trigonometric ratio

    sin⁡π6=12\sin \frac{\pi}{6}=\frac{1}{2}

    Use the special triangles or the unit circle for this exact ratio.

  3. State the exact value

    csc⁡π6=2\csc \frac{\pi}{6}=2

    This is the required exact value.

Answer
22
Question 3
2 markseasy
Find the exact value of cot⁡π4\cot \frac{\pi}{4}.

Worked solution

  1. Recall the definition of the reciprocal function

    cot⁡x=1tan⁡x\cot x=\dfrac{1}{\tan x}

    The reciprocal function is one divided by the matching basic ratio.

  2. Evaluate the underlying trigonometric ratio

    tan⁡π4=1\tan \frac{\pi}{4}=1

    Use the special triangles or the unit circle for this exact ratio.

  3. State the exact value

    cot⁡π4=1\cot \frac{\pi}{4}=1

    This is the required exact value.

Answer
11
Question 4
2 markseasy
Find the exact value of sec⁡π4\sec \frac{\pi}{4}.

Worked solution

  1. Recall the definition of the reciprocal function

    sec⁡x=1cos⁡x\sec x=\dfrac{1}{\cos x}

    The reciprocal function is one divided by the matching basic ratio.

  2. Evaluate the underlying trigonometric ratio

    cos⁡π4=22\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}

    Use the special triangles or the unit circle for this exact ratio.

  3. State the exact value

    sec⁡π4=2\sec \frac{\pi}{4}=\sqrt{2}

    This is the required exact value.

Answer
2\sqrt{2}
Question 5
2 markseasy
Find the exact value of arcsin⁡(12)\arcsin\left(\frac{1}{2}\right), giving your answer in radians.

Worked solution

  1. Rewrite using the forward trigonometric function

    θ=arcsin⁡(12) ⇒ sin⁡θ=12\theta=\arcsin\left(\frac{1}{2}\right)\ \Rightarrow\ \sin\theta=\frac{1}{2}

    The inverse function asks: which angle has this ratio?

  2. State the principal value range

    θ∈[−π2,π2]\theta\in \left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]

    The answer must lie in the restricted range of the inverse function.

  3. State the inverse value

    arcsin⁡(12)=π6\arcsin\left(\frac{1}{2}\right)=\frac{\pi}{6}

    This angle lies in the correct range.

Answer
π6\frac{\pi}{6}

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