IB Mathematics Analysis and Approaches Topic SL…

Consider the equation

3\cos(2x)=1-\sin x, \qquad 0\le x<2\pi.

(a) Show that the equation may be written as:

6\cos^2x+\sin x-4=0.

(b) Express $\cos^2x$ in terms of $\sin x$ , and hence obtain a quadratic equation in $\sin x$ .

The depth, $h(t)$ metres, of water at the entrance to a harbour at time $t$ hours after midnight on a particular day is given by $h(t)=8+4 \sin \left(\frac{\pi t}{6}\right)$ , $0 \leq t \leq 24$ .

(a) Find the maximum depth and the minimum depth of the water.

(b) Find the values of $t$ for which $h(t) \geq 8$

Consider the equation $\cos(2x)=\cos x$ , $0\le x<2\pi$ .

(a) Show that the equation may be written as $2\cos^2 x-\cos x-1=0$ .

(b) Factor the left-hand side to obtain an equation of the form $(2\cos x+1)(\cos x-1)=0$ .

Consider the equation $\cos(2x)+3\sin x=-1$ , $0\le x<2\pi$ .

(a) Show that the equation may be written as $\cos^2 x+\frac{3}{2}\sin x=0$ .

(b) Express $\cos^2 x$ in terms of $\sin x$ , and hence obtain a quadratic equation in $\sin x$ .

The graph of $y = \cos x$ is transformed by the following sequence of four transformations.

A horizontal stretch with scale factor $0.8$ , then a translation of $1.6$ units in the positive $x$ -direction.

A vertical stretch with scale factor $2.5$ , then a translation of $3.1$ units in the positive $y$ -direction.

(a) Find the equation of the resulting graph in the form $y = a\cos\left(b(x+c)\right) + d$ .

(b) Write down the period of this graph and the range of the corresponding function.

(d) Find the smallest positive value of $x$ at which this graph meets the line $y = x$ .

(d) all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the original equation.

(e) the graphs of $y=3\cos(2x)$ and $y=1-\sin x$ for $0\le x\le2\pi$ , indicating approximate intersections matching the solutions from part 4.

(d) all exact values of $x$ in the interval $0\le x<2\pi$ that satisfy the original equation.

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Questionbank

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Questionbank

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Questionbank

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 3.7—circular Functions, Graphs, Composites, Transformations Questionbank