Secant, cosecant and cotangentEdexcel International A Level Maths: Subtopic test

1. 1
   The angle $\theta$ is acute and $\sin\theta=\frac{5}{13}$.

   (a)

   Find the value of $\operatorname{cosec}\,\theta$.

   [1 mark]

   • A$\frac{5}{13}$
   • B$\frac{13}{5}$
   • C$\frac{13}{12}$
   • D$\frac{12}{5}$

   (b)

   Find the value of $\cot\theta$.

   [1 mark]

   • A$\frac{5}{12}$
   • B$\frac{13}{12}$
   • C$\frac{13}{5}$
   • D$\frac{12}{5}$

   (c)

   Find the exact value of $\sec\theta+\operatorname{cosec}\,\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Throughout, $x$ is measured in radians, and each function is considered only where it is defined.

   (a)

   For which values of $x$ is $\sec x$ undefined?

   [1 mark]

   • A$x=\frac{\pi}{2}+n\pi$, where $n$ is an integer
   • B$x=n\pi$, where $n$ is an integer
   • C$x=2n\pi$, where $n$ is an integer
   • D$x=\frac{n\pi}{2}$, where $n$ is an integer

   (b)

   What is the range of $\operatorname{cosec}\,x$?

   [1 mark]

   • A$-1\le y\le1$
   • Ball real numbers
   • C$y\le-1$ or $y\ge1$
   • D$y\ge1$

   (c)

   Write $\dfrac{\operatorname{cosec}\,x}{\cot x}$ as a single trigonometric function.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are measured in degrees and $0\le\theta<360$.

   (a)

   Solve $\sec\theta=-2$.

   [3 marks]

   (b)

   Solve $\sec\theta=3\cos\theta$, giving your answers to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Throughout, $x$ is measured in radians with $0<x<\pi$. A calculator may be used.

   (a)

   The function $f$ is defined by $f(x)=2\operatorname{cosec}\,x+1$.
   (i) State the minimum value of $f$ and the value of $x$ at which it occurs.
   (ii) Solve $f(x)=5$, giving exact answers.
   (iii) Explain why $f(x)=2$ has no solutions.

   [6 marks]

   (b)

   (i) Show that the equation $\cot x=2\sin x$ can be written as $2\cos^2x+\cos x-2=0$.
   (ii) Hence solve $\cot x=2\sin x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions