Test of Mathematics for University Admission (TMUA) Trigonometry Flashcards

24 sample cards from the Trigonometry deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Describe the transformation taking $y = f(x)$ to $y = f(x - c)$ and to $y = f(x) + d$.

   $y = f(x - c)$ is a translation by $c$ in the positive $x$-direction (right). $y = f(x) + d$ is a translation by $d$ in the positive $y$-direction (up).

2. What transformations map $y = \cos x$ onto $y = \cos(-x)$ and onto $y = -\cos x$?

   $y = \cos(-x) = \cos x$ (reflection in the $y$-axis leaves it unchanged, as cosine is even). $y = -\cos x$ is a reflection in the $x$-axis.

3. For $y = a\sin(bx + c) + d$, identify the amplitude, period and vertical shift.

   Amplitude $= |a|$, period $= \dfrac{2\pi}{|b|}$, and vertical shift (midline) $= d$ so the range is $d - |a| \leq y \leq d + |a|$.

4. State the Pythagorean identity relating $\sin$ and $\cos$.

   $\sin^{2}\theta + \cos^{2}\theta = 1$.

5. Rearrange $\sin^{2}\theta + \cos^{2}\theta = 1$ to express $\sin^{2}\theta$ and $\cos^{2}\theta$ individually.

   $\sin^{2}\theta = 1 - \cos^{2}\theta$ and $\cos^{2}\theta = 1 - \sin^{2}\theta$.

6. What is the identity for $\tan\theta$ in terms of $\sin\theta$ and $\cos\theta$, and where is it valid?

   $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, valid for $\cos\theta \neq 0$, i.e. $\theta \neq \tfrac{\pi}{2} + n\pi$.

7. In which quadrants are $\sin$, $\cos$ and $\tan$ positive (the CAST rule)?

   All positive in quadrant 1; only $\sin$ positive in quadrant 2; only $\tan$ positive in quadrant 3; only $\cos$ positive in quadrant 4 (remembered as CAST going anticlockwise from Q4).

8. Give the symmetry identities relating $\sin(180^{\circ} - \theta)$ and $\cos(180^{\circ} - \theta)$ to $\theta$.

   $\sin(180^{\circ} - \theta) = \sin\theta$ and $\cos(180^{\circ} - \theta) = -\cos\theta$.

9. State the complementary-angle identities relating sine and cosine.

   $\sin(90^{\circ} - \theta) = \cos\theta$ and $\cos(90^{\circ} - \theta) = \sin\theta$ (equivalently $\sin\left(\tfrac{\pi}{2} - \theta\right) = \cos\theta$).

10. When solving $\sin\theta = k$ for $0^{\circ} \leq \theta < 360^{\circ}$, how do you find the second solution from the principal value $\alpha$?

    The two solutions are $\theta = \alpha$ and $\theta = 180^{\circ} - \alpha$ (then add multiples of $360^{\circ}$ to find others in range).

11. When solving $\cos\theta = k$ for $0^{\circ} \leq \theta < 360^{\circ}$, how do you find the second solution from the principal value $\alpha$?

    The solutions are $\theta = \alpha$ and $\theta = 360^{\circ} - \alpha$ (i.e. $\pm\alpha$ plus multiples of $360^{\circ}$).

12. When solving $\tan\theta = k$, how are successive solutions related?

    Solutions repeat every $180^{\circ}$ (or $\pi$): if $\theta = \alpha$ is one solution, then $\theta = \alpha + 180^{\circ}n$ for integer $n$.

13. Outline the process for solving an equation such as $\sin(2x) = k$ over a given interval for $x$.

    Multiply the interval for $x$ by the coefficient to get the interval for the argument $u = 2x$, solve $\sin u = k$ for all $u$ in that wider range, then divide each solution by $2$ to recover $x$.

14. How can $2\cos^{2}\theta + \cos\theta - 1 = 0$ be solved using an identity or substitution?

    It is already quadratic in $\cos\theta$: let $c = \cos\theta$, solve $2c^{2} + c - 1 = 0$ to get $c = \tfrac{1}{2}$ or $c = -1$, then solve $\cos\theta = \tfrac{1}{2}$ and $\cos\theta = -1$.

15. How do you solve an equation mixing $\sin^{2}\theta$ and $\cos\theta$, such as $2\sin^{2}\theta = 1 + \cos\theta$?

    Use $\sin^{2}\theta = 1 - \cos^{2}\theta$ to write everything in $\cos\theta$, giving a quadratic in $\cos\theta$, then solve.

16. State the compound-angle formulae for $\sin(A \pm B)$.

    $\sin(A + B) = \sin A\cos B + \cos A\sin B$ and $\sin(A - B) = \sin A\cos B - \cos A\sin B$.

17. State the compound-angle formulae for $\cos(A \pm B)$.

    $\cos(A + B) = \cos A\cos B - \sin A\sin B$ and $\cos(A - B) = \cos A\cos B + \sin A\sin B$. (Note the sign 'swap' relative to the sine formula.)

18. State the compound-angle formula for $\tan(A \pm B)$.

    $\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$.

19. State the double-angle formula for $\sin 2A$.

    $\sin 2A = 2\sin A\cos A$.

20. State the three equivalent forms of the double-angle formula for $\cos 2A$.

    $\cos 2A = \cos^{2}A - \sin^{2}A = 2\cos^{2}A - 1 = 1 - 2\sin^{2}A$.

21. State the double-angle formula for $\tan 2A$.

    $\tan 2A = \dfrac{2\tan A}{1 - \tan^{2}A}$.

22. How can $\cos 2A = 2\cos^{2}A - 1$ be rearranged to give $\cos^{2}A$ (the power-reduction form)?

    $\cos^{2}A = \dfrac{1 + \cos 2A}{2}$, and similarly $\sin^{2}A = \dfrac{1 - \cos 2A}{2}$.

23. Using a double-angle identity, what does $\sin\theta\cos\theta$ simplify to?

    $\sin\theta\cos\theta = \dfrac{1}{2}\sin 2\theta$ (from $\sin 2\theta = 2\sin\theta\cos\theta$).

24. Why must you check candidate solutions of a trigonometric equation against the stated interval and any domain restrictions?

    Squaring, dividing by a trig term, or using identities can introduce extraneous solutions or lose solutions (e.g. where $\cos\theta = 0$ for $\tan$); each candidate must lie in the given interval and satisfy the original equation.

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Trigonometry flashcards FAQ