Methods2025 NHT Exam 1 · Q3Tech-free7 marks

Tangent and sine functions: solutions, intersections and sketch

Let $f : D \to \R, f(x) = \frac{1}{2}\tan(x) + \frac{1}{2}$ , where is the maximal domain.

Find the general solution for $f(x) = 0$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

b.i.

Let $g : \R \to \R, g(x) = \sin(x) + \frac{1}{2}$ .

Find all values of $x$ such that $f(x) = g(x)$ for $x \in (-\pi, \pi)$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

b.ii.

The graph of $y = g(x)$ is shown below.

On the set of axes below, sketch the graph of $y = f(x)$ over the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , labelling all points where $f(x) = g(x)$ with their coordinates and all asymptotes with their equations.

Grid with axes marked at x = -π/2 and π/2 and y = -1, 1 and 2. It shows the graph of y = g(x) = sin(x) + 1/2, with a minimum of -1/2 at x = -π/2, a y-intercept of 1/2 and a maximum of 3/2 at x = π/2. Students sketch y = f(x) on these axes.

3 marks

Hints, answer checking and worked solutions come with a Season Pass.