Analytic trigonometry: introduction

Before starting the unit, measure your starting point. It is a short self-assessment: it is not graded and its result is stored only in your browser.

1. Question 1Which of the following equalities is an identity on its common domain?

   $\sec x - \cos x = \sin x \tan x$

   $\sqrt{1 - \cos^2 x} = \sin x$

   $\sin x + \cos x = 1$

   $\frac{\tan x}{\sin x} = \cos x$

2. Question 2If $\tan \theta = -\frac{3}{4}$ and $\theta$ lies in the second quadrant, calculate $\cos \theta$.

3. Question 3Calculate $\tan \frac{7\pi}{12}$, writing $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$.

   $-2 - \sqrt{3}$

   $1 + \sqrt{3}$

   $2 + \sqrt{3}$

   $2 - \sqrt{3}$

4. Question 4If $\sin \theta = \frac{3}{5}$ and $\frac{\pi}{2} < \theta < \pi$, calculate $\sin 2\theta$.

5. Question 5Calculate the maximum value of $f(x) = 3 \sin x - 4 \cos x + 1$.

6. Question 6How many solutions does the equation $2 \cos^2 x - \cos x - 1 = 0$ have in $\ico{0}{2\pi}$?

7. Question 7Solving $\cos x - \sin x = 1$ by squaring yields the candidates $x = \frac{k\pi}{2}$. Which are the solutions in $\ico{0}{2\pi}$?

   $0$ and $\frac{3\pi}{2}$

   $0$, $\frac{\pi}{2}$, $\pi$ and $\frac{3\pi}{2}$

   $0$ and $\frac{\pi}{2}$

   $\frac{\pi}{2}$ and $\pi$

8. Question 8What angle do the lines $y = 2x + 1$ and $y = -3x + 4$ form?

   $\frac{\pi}{4}$

   $\frac{3\pi}{4}$

   $\arctan 5$

   $\frac{\pi}{2}$

9. Question 9What does the equation $x^2 + 4xy + y^2 - 3 = 0$ represent?

   A hyperbola.

   An ellipse.

   A circle.

   A parabola.

10. Question 10From a log of circular cross section, $20$ centimeters in diameter, a beam of rectangular cross section is to be cut, with the four vertices of the cross section on the circle of the log. If $\theta$ is the angle that a diagonal of the rectangle forms with one of its sides, what is the largest possible area of the cross section, in square centimeters?

11. Question 11From earlier unitsWhat are the coordinates of the point $P\left( -\frac{7\pi}{6} \right)$ on the unit circle?

    $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$

    $\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$

    $\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$

    $\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$

12. Question 12From earlier unitsCalculate $\arccos\left( \cos \frac{4\pi}{3} \right)$.

    $\frac{2\pi}{3}$

    $\frac{4\pi}{3}$

    $-\frac{2\pi}{3}$

    $\frac{\pi}{3}$

13. Question 13From earlier unitsCalculate the cosine of the angle formed by the vectors $\left( 3, 4 \right)$ and $\left( 4, 3 \right)$.