Signs, values and periods

Signs in the four quadrants

The two axes cut the plane into four quadrants. The sign of $\cos x$ is the sign of the x-coordinate of $P$, and the sign of $\sin x$ is the sign of its y-coordinate. The other four functions take their signs from these two.

Cosec has the sign of sin, sec has the sign of cos, and cot has the sign of tan.

A short way to remember the positive ones, going round from quadrant I, is All, Sin, Tan, Cos.

The values to know

These come from the triangles of Class 10: half a square for $\dfrac{\pi}{4}$, and half of an equilateral triangle for $\dfrac{\pi}{6}$ and $\dfrac{\pi}{3}$.

Angles in the other quadrants

The circle is symmetric, and that gives every other value.

• The point at $\pi - x$ is the mirror image of the point at $x$ in the y-axis. It has the same height, and the opposite x-coordinate.
• The point at $\pi + x$ is opposite $P$. Both of its coordinates have the opposite sign.
• The point at $2\pi - x$, which is the same as $-x$, is the mirror image of $P$ in the x-axis.

The method is always the same. Find the acute angle that the radius $OP$ makes with the x-axis. Take the value for that acute angle, and give it the sign of the quadrant. For example:

• $\sin \dfrac{5\pi}{6} = \sin \dfrac{\pi}{6} = \dfrac{1}{2}$, since sine is positive in quadrant II;
• $\cos \dfrac{7\pi}{6} = -\cos \dfrac{\pi}{6} = -\dfrac{\sqrt{3}}{2}$, since cosine is negative in quadrant III;
• $\tan \dfrac{11\pi}{6} = -\dfrac{1}{\sqrt{3}}$, since tangent is negative in quadrant IV.

The mirror image in the x-axis also gives $\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$. Cosine is an even function, and sine is an odd function.

Set $k = 5$, then $k = 7$, then $k = 11$. The grey point stays at the same place, so the sizes stay the same. Only the signs change.

Whole turns change nothing

After a full turn of $2\pi$ the point is back where it started. So for every integer $n$,

To find a value at a large angle, take away whole turns first. For example, $\sin \dfrac{31\pi}{3} = \sin\left(10\pi + \dfrac{\pi}{3}\right) = \sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$, because $10\pi$ is five whole turns.

Periods

A function $f$ is periodic if $f(x + T) = f(x)$ for every $x$, for some $T > 0$. The smallest such $T$ is the period.

• Sine, cosine, secant and cosecant have period $2\pi$.
• Tangent and cotangent have period $\pi$. The point at $x + \pi$ is opposite $P$, so both coordinates change sign and their ratio stays the same: $\tan(x + \pi) = \tan x$.

Domains and ranges

• Sine and cosine. The domain is all real numbers, and the range is $[-1, 1]$.
• Tangent and secant. The domain is all real $x$ except the odd multiples of $\dfrac{\pi}{2}$.
• Cotangent and cosecant. The domain is all real $x$ except the multiples of $\pi$.
• Tangent and cotangent take every real value.
• Secant and cosecant take the values in $(-\infty, -1] \cup [1, \infty)$. A number of size at most $1$ has a reciprocal of size at least $1$.

How each function changes

As $x$ runs from $0$ to $2\pi$, $\sin x$ rises from $0$ to $1$, falls through $0$ to $-1$, and rises back to $0$. $\cos x$ starts at $1$, falls to $-1$ at $\pi$, and rises back to $1$ at $2\pi$.

Practise with new numbers

Try it in Python

This program takes away whole turns from four angles, finds each quadrant, and prints the three values with their signs.

Common mistakes

• Taking away half turns. For sine and cosine, take away multiples of $2\pi$, not of $\pi$. Only tan and cot repeat every $\pi$.
• The wrong sign. $\cos \dfrac{2\pi}{3}$ is $-\dfrac{1}{2}$, not $\dfrac{1}{2}$, because the angle is in quadrant II.
• Making cosine odd. $\cos(-x) = \cos x$. It is sine that changes sign.
• The range of secant. It is not $[-1, 1]$. It is everything outside the open interval $(-1, 1)$.

Remember these

• Positive in quadrants I, II, III, IV: all; sin and cosec; tan and cot; cos and sec.
• Value at any angle: take away whole turns, find the acute angle with the x-axis, then give the sign of the quadrant.
• $\cos(-x) = \cos x$ and $\sin(-x) = -\sin x$.
• Periods: $2\pi$ for sin, cos, sec and cosec; $\pi$ for tan and cot.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In which quadrant is $x$ if $\sin x < 0$ and $\cos x > 0$?

2. Find $\cos \dfrac{19\pi}{3}$.

3. Find $\tan\left(-\dfrac{3\pi}{4}\right)$.

4. The cotangent repeats itself. Why is its period $\pi$ and not $2\pi$?

5. Can $\sec x = \dfrac{1}{2}$ for some $x$?