Solving trigonometric equationsEdexcel International A Level Maths: Subtopic test

1. 1
   $\sin\theta=0.6$, for $0^\circ\le\theta\le360^\circ$.

   (a)

   Which of the following is the larger of the two solutions?

   [1 mark]

   • A$36.9^\circ$
   • B$143.1^\circ$
   • C$216.9^\circ$
   • D$323.1^\circ$

   (b)

   Find the sum of the two solutions.

   [1 mark]

   • A$180^\circ$
   • B$360^\circ$
   • C$90^\circ$
   • D$36.9^\circ$

   (c)

   Hence solve $\sin(\theta-20^\circ)=0.6$ for $0^\circ\le\theta\le360^\circ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In this question, $0\le x<2\pi$ and angles are in radians.

   (a)

   Solve $\cos x=-\frac12$.

   [1 mark]

   • A$\frac\pi3$ and $\frac{5\pi}{3}$
   • B$\frac{2\pi}{3}$ and $\frac{5\pi}{3}$
   • C$\frac{2\pi}{3}$ and $\frac{4\pi}{3}$
   • D$\frac\pi3$ and $\frac{2\pi}{3}$

   (b)

   How many solutions does $\tan2x=1$ have?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Solve $\sin\left(x+\frac\pi2\right)=\frac34$, giving your answers to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Give angles in degrees, with non-exact answers to 1 decimal place.

   (a)

   Solve $\cos(x+30^\circ)=\frac12$ for $-180^\circ<x<180^\circ$.

   [3 marks]

   (b)

   Solve $6\cos^2x+\sin x-5=0$ for $0^\circ\le x<360^\circ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water $D$ metres in a harbour, $t$ hours after midnight with $0\le t<12$, is modelled in two ways, using $\theta=\frac{\pi t}{6}$ radians. Model A: $D=5+2\sin\theta$. Model B: $D=2\sin^2\theta+\cos\theta+1$.

   (a)

   Using Model A, find the times in the interval $0\le t<12$ when
   (i) the depth is $6$ m,
   (ii) the depth is $4$ m.

   [6 marks]

   (b)

   Using Model B, find the times in the interval $0\le t<12$ when the depth is $3$ m.

   [6 marks]

   Total for question 4: 12 marks

End of questions