Question4:
(a) Diagram 2 shows the graph of y = tanx + c for 0 ⩽ x ⩽ p, such that c and p are constants.
State the value of c and of p. [2 marks]
Solution:
(a)
$$ \begin{aligned} & y=\tan x+c \\ & c=-1, \quad \text { graph moves down } 1 \text { unit } \\ & \text { At }(p,-2) \\ & -2=\tan p-1 \\ & \tan p=-1 \text { (quadrant II and IV) } \end{aligned} $$
$$ \begin{aligned} \text { Reference angle } & =\tan ^{-1}(1) \\ & =45^{\circ} \\ \therefore p & =360^{\circ}-45^{\circ} \\ p & =315^{\circ} \times \frac{\pi}{180^{\circ}} \\ p & =\frac{7 \pi}{4}(\text { quadrant } I V) \end{aligned} $$
(b)(i)
$$ \begin{aligned} & 2 \text { cos } x=-1 ; \pi \leqslant x \leqslant 3 \pi \\ & \begin{aligned} \text { cos } x=-\frac{1}{2}(\text { quadrant II and III }) \end{aligned} \\ & \begin{aligned} \text { Reference angle } & =\text { cos }^{-1}\left(\frac{1}{2}\right) \\ & =60^{\circ} \\ & =\frac{\pi}{3} \end{aligned} \end{aligned} $$
$$ \begin{aligned} & x=\pi-\frac{\pi}{3}, \quad \pi+\frac{\pi}{3}, \quad 3 \pi-\frac{\pi}{3} \\ & x=\frac{2 \pi}{3}, \quad \frac{4 \pi}{3}, \quad \frac{8 \pi}{3} \\ & \therefore x=\frac{4 \pi}{3}, \quad \frac{8 \pi}{3} \end{aligned} $$
(b)(ii)
$$ \begin{aligned} & y=2 \operatorname{cos} x \text { for } 0 \leqslant x \leqslant 2 \pi \\ & \operatorname{cos} x<-\frac{1}{2} \\ & 2 \operatorname{cos} x<-1 \\ & \qquad y<-1 \end{aligned} $$
(a) Diagram 2 shows the graph of y = tanx + c for 0 ⩽ x ⩽ p, such that c and p are constants.
State the value of c and of p. [2 marks]

(b) For the following questions, give your answer in the simplest fraction form in terms of π radians.
(i) Solve the equation 2 cos x = -1 for π ⩽ x ⩽ 3π.
(ii) Sketch the graph of y = 2 cos x for 0 ⩽ x ⩽ 2π. By drawing a suitable straight line on your sketch, find the range of values of x for which cos x < -1/2 for 0 ⩽ x ⩽ 2π.
[6 marks]
Solution:
(a)
$$ \begin{aligned} & y=\tan x+c \\ & c=-1, \quad \text { graph moves down } 1 \text { unit } \\ & \text { At }(p,-2) \\ & -2=\tan p-1 \\ & \tan p=-1 \text { (quadrant II and IV) } \end{aligned} $$
$$ \begin{aligned} \text { Reference angle } & =\tan ^{-1}(1) \\ & =45^{\circ} \\ \therefore p & =360^{\circ}-45^{\circ} \\ p & =315^{\circ} \times \frac{\pi}{180^{\circ}} \\ p & =\frac{7 \pi}{4}(\text { quadrant } I V) \end{aligned} $$
(b)(i)
$$ \begin{aligned} & 2 \text { cos } x=-1 ; \pi \leqslant x \leqslant 3 \pi \\ & \begin{aligned} \text { cos } x=-\frac{1}{2}(\text { quadrant II and III }) \end{aligned} \\ & \begin{aligned} \text { Reference angle } & =\text { cos }^{-1}\left(\frac{1}{2}\right) \\ & =60^{\circ} \\ & =\frac{\pi}{3} \end{aligned} \end{aligned} $$
$$ \begin{aligned} & x=\pi-\frac{\pi}{3}, \quad \pi+\frac{\pi}{3}, \quad 3 \pi-\frac{\pi}{3} \\ & x=\frac{2 \pi}{3}, \quad \frac{4 \pi}{3}, \quad \frac{8 \pi}{3} \\ & \therefore x=\frac{4 \pi}{3}, \quad \frac{8 \pi}{3} \end{aligned} $$
(b)(ii)
$$ \begin{aligned} & y=2 \operatorname{cos} x \text { for } 0 \leqslant x \leqslant 2 \pi \\ & \operatorname{cos} x<-\frac{1}{2} \\ & 2 \operatorname{cos} x<-1 \\ & \qquad y<-1 \end{aligned} $$

From the working in (b)(i) , 2 cos x = -1, range of values of x for 0 ⩽ x ⩽ 2π is 2π/3 ⩽ x ⩽ 4π/3.