1. 1
   An engineer is analysing a signal modelled by $y = \sin x$ for $0° \leq x \leq 360°$.

   (a)

   For what value of x does $\sin x$ first reach its maximum value?

   [1 mark]

   • A90 degrees
   • B0 degrees
   • C180 degrees
   • D270 degrees

   (b)

   How many solutions does the equation $\sin x = 0.5$ have in the range $0° \leq x \leq 360°$?

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (c)

   What is the value of $\cos x$ when $x = 180°$?

   [1 mark]

   • A0.5
   • B0
   • C1
   • D-1

   Total for question 1: 3 marks

2. 2
   A student is solving the equation $2\cos x + 1 = 0$ for $0° \leq x \leq 360°$.

   (a)

   Rearrange to find the value of $\cos x$, and find the value of x in the range $90° \leq x \leq 180°$ that satisfies the equation.

   [2 marks]

   (b)

   Using the symmetry of the cosine graph, find the other value of x in the range $0° \leq x \leq 360°$ that satisfies $\cos x = -0.5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A researcher is studying oscillations modelled by $y = \tan x$.

   (a)

   Find the value of x in the range $0° \leq x \leq 90°$ that satisfies $\tan x = \sqrt{3}$.

   [3 marks]

   (b)

   Hence find the other value of x in the range $0° \leq x \leq 360°$ that satisfies $\tan x = \sqrt{3}$, explaining why $\tan x$ is positive there as well.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A physicist models a wave using $y = \sin x$ for $0° \leq x \leq 360°$.

   (a)

   Solve the equation $\sin x = 0.643$, giving both solutions in the range correct to 1 decimal place.

   [4 marks]

   (b)

   Given that $\sin x$ reaches a maximum value of 1 at $x = 90°$, and using your answers from part (a), describe how $\sin x$ changes as x increases from $40.0°$ to $140.0°$, and state the maximum value it reaches in this interval and where it occurs.

   [4 marks]

   (c)

   The physicist now wants to solve $2\sin x - 1 = 0$ for $0° \leq x \leq 360°$ and compare the number of solutions to part (a). Solve the equation, giving all solutions correct to 1 decimal place, and state whether this equation has the same number of solutions in the range as $\sin x = 0.643$ from part (a).

   [5 marks]

   Total for question 4: 13 marks

End of questions