Where does y=\tan x have vertical asymptotes?

At x=\frac{\pi}{2}+n\pi, where \cos x=0

Exact values of \sin at 0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}?

0,\ \frac12,\ \frac{\sqrt2}{2},\ \frac{\sqrt3}{2},\ 1

Exact values of \cos at 0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}?

1,\ \frac{\sqrt3}{2},\ \frac{\sqrt2}{2},\ \frac12,\ 0

Exact values of \tan at 0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3}?

0,\ \frac{1}{\sqrt3},\ 1,\ \sqrt3

Undefined, because \cos\frac{\pi}{2}=0

Exact value of \cos\frac{2\pi}{3}?

-\frac12 (reference angle \frac{\pi}{3}, cosine negative in the second quadrant)

If \sin x=k has a solution \alpha in [0,2\pi], what is the other?

\pi-\alpha (reflection in x=\frac{\pi}{2})

How do you find \sin\frac{7\pi}{6} exactly?

Reference angle \frac{\pi}{6} gives \frac12; sine is negative in the third quadrant, so -\frac12

Trigonometric graphs and exact valuesAQA A-Level Maths: Flashcards

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Period of $\sin x$ and $\cos x$?

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Period of $\sin x$ and $\cos x$ ?: $2\pi$ radians ( $360^\circ$ )
Period of $\tan x$ ?: $\pi$ radians ( $180^\circ$ )
Range of $\sin x$ and $\cos x$ ?: $-1\le y\le1$
Where does $y=\tan x$ have vertical asymptotes?: At $x=\frac{\pi}{2}+n\pi$ , where $\cos x=0$
$\sin(-x)$ , $\cos(-x)$ , $\tan(-x)$ ?: $-\sin x$ , $\cos x$ , $-\tan x$
$\sin(\pi-x)$ and $\cos(\pi-x)$ ?: $\sin x$ and $-\cos x$
Exact values of $\sin$ at $0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$ ?: $0,\ \frac12,\ \frac{\sqrt2}{2},\ \frac{\sqrt3}{2},\ 1$
Exact values of $\cos$ at $0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$ ?: $1,\ \frac{\sqrt3}{2},\ \frac{\sqrt2}{2},\ \frac12,\ 0$
Exact values of $\tan$ at $0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3}$ ?: $0,\ \frac{1}{\sqrt3},\ 1,\ \sqrt3$
$\tan\frac{\pi}{2}$ ?: Undefined, because $\cos\frac{\pi}{2}=0$
Exact value of $\cos\frac{2\pi}{3}$ ?: $-\frac12$ (reference angle $\frac{\pi}{3}$ , cosine negative in the second quadrant)
If $\sin x=k$ has a solution $\alpha$ in $[0,2\pi]$ , what is the other?: $\pi-\alpha$ (reflection in $x=\frac{\pi}{2}$ )
How do you find $\sin\frac{7\pi}{6}$ exactly?: Reference angle $\frac{\pi}{6}$ gives $\frac12$ ; sine is negative in the third quadrant, so $-\frac12$

Exam questions on Trigonometric graphs and exact values

The curve $y=\sin x$ is drawn for $0\le x\le2\pi$ , where $x$ is in radians.
Use the symmetry of the graph to find both solutions of $\sin x=-\frac12$ in the interval, in terms of $\pi$ .2 marks
The function $g(x)=\tan x$ is considered for $x$ in degrees.
Given that $\tan40^\circ=t$ , write down in terms of $t$ (i) $\tan140^\circ$ , (ii) $\tan220^\circ$ .2 marks
The points $P$ and $Q$ lie on the curve $y=\cos x$ , where $x$ is in radians. The $x$ -coordinate of $P$ is $\frac{\pi}{3}$ and the $x$ -coordinate of $Q$ is $\frac{2\pi}{3}$ .
Find the exact distance $PQ$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).