AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically Questionbank

Write down the coordinates of $Q$.

Find the value of $\tan \theta$ at $\theta=\frac{7 \pi}{6}$, giving your answer in exact form.

Another point $R$ lies on the unit circle such that $\sin \phi=-\frac{1}{2}$ and $\phi \in(0,2 \pi)$. Find all possible values of $\phi$.

Solve $\tan x = 1$ for $x$ in the interval $0 \leq x < 2\pi$.

Use a graphing calculator to confirm your solution.

Explain why there are exactly two solutions for $\tan x = 1$ in this interval.

Describe the behaviour of the tangent function near its asymptotes.

Define $\cos \theta$ and $\sin \theta$ in terms of the unit circle.

Prove the Pythagorean identity:

$\cos^2 \theta + \sin^2 \theta = 1$

Given that $\sin \theta = \frac{3}{5}$ for some acute angle $\theta$, find $\cos \theta$.

Use the above result to determine $\tan \theta$.

Explain the significance of the unit circle in constructing the graphs of ${f(x) = \sin x}$ and ${f(x) = \cos x}$.

Solve $\sin x = \frac{1}{2}$ for $x$ in the interval $0 \leq x < 2\pi$.

Graphically interpret the solutions of $\sin x = \frac{1}{2}$

Find the phase shift and period of the function:

$f(x) = \cos(x - \frac{\pi}{4})$

Verify that $\tan \left(\frac{\pi}{4}\right) = 1$

How can graphical methods help in solving trigonometric equations within a finite interval?

Sketch the graph of $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$.

Using your graph, estimate the solutions to the equation $2\sin(x) - \cos(x) = 0$ within the interval $0 \leq x \leq 2\pi$.