Trigonometric graphs and exact valuesAQA A-Level Maths: Subtopic test

1. 1
   The curve $y=\sin x$ is drawn for $0\le x\le2\pi$, where $x$ is in radians.

   (a)

   Find the exact value of $\sin\frac{5\pi}{6}$.

   [1 mark]

   • A$\frac{\sqrt3}{2}$
   • B$\frac12$
   • C$-\frac12$
   • D$-\frac{\sqrt3}{2}$

   (b)

   The line $y=\frac{\sqrt3}{2}$ meets the curve at two points. Find the $x$-coordinate of the second point (the one with the larger $x$).

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{5\pi}{6}$
   • C$\frac{4\pi}{3}$
   • D$\frac{2\pi}{3}$

   (c)

   Use the symmetry of the graph to find both solutions of $\sin x=-\frac12$ in the interval, in terms of $\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g(x)=\tan x$ is considered for $x$ in degrees.

   (a)

   Find the exact value of $\tan120^\circ$.

   [1 mark]

   • A$-\sqrt3$
   • B$\sqrt3$
   • C$-\frac{1}{\sqrt3}$
   • D$\frac{1}{\sqrt3}$

   (b)

   Which statement about the graph of $y=\tan x$ is correct?

   [1 mark]

   • AIt is bounded between $-1$ and $1$.
   • BIt has period $360^\circ$.
   • CIt has a vertical asymptote at $x=90^\circ$ and repeats every $180^\circ$.
   • DIt passes through the point $(90^\circ,1)$.

   (c)

   Given that $\tan40^\circ=t$, write down in terms of $t$ (i) $\tan140^\circ$, (ii) $\tan220^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $P$ and $Q$ lie on the curve $y=\cos x$, where $x$ is in radians. The $x$-coordinate of $P$ is $\frac{\pi}{3}$ and the $x$-coordinate of $Q$ is $\frac{2\pi}{3}$.

   (a)

   Find the exact distance $PQ$.

   [3 marks]

   (b)

   The straight line through $P$ and $Q$ meets the $x$-axis at $R$. Find the exact $x$-coordinate of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curves $y=\sin x$ and $y=\cos x$ are drawn for $0\le x\le2\pi$, where $x$ is in radians. No calculator may be used.

   (a)

   (i) Show that the curves intersect at $x=\frac{\pi}{4}$ and state the exact $y$-coordinate.
   (ii) Show that they also intersect at $x=\frac{5\pi}{4}$ and state the exact $y$-coordinate.
   (iii) Both curves repeat every $2\pi$. Find, with justification, the number of intersections for $0\le x\le6\pi$.

   [6 marks]

   (b)

   Let $d=\sin x-\cos x$.
   (i) Find the exact value of $d$ when $x=\frac{3\pi}{4}$.
   (ii) Find the exact value of $d$ when $x=\frac{7\pi}{4}$.
   (iii) Hence explain why the curves must meet at a value of $x$ between $\frac{3\pi}{4}$ and $\frac{7\pi}{4}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions