Trigonometric Functions and Their Graphs: Quick Revision

A brief note on Trigonometric Functions and Their Graphs. Read it before a test. Try each question at the end before you open its answer.

The unit circle

• Turn the point $A(1, 0)$ through an angle $x$ on the circle of radius $1$. The new point is $P = (\cos x, \sin x)$.
• This defines sine and cosine for every real $x$. Both lie between $-1$ and $1$.
• $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$ and $1 + \cot^2 x = \operatorname{cosec}^2 x$.
• $\tan x = \dfrac{\sin x}{\cos x}$ and $\cot x = \dfrac{\cos x}{\sin x}$.
• $\sec x = \dfrac{1}{\cos x}$ and $\operatorname{cosec} x = \dfrac{1}{\sin x}$.

Signs and values

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

Periods, domains and ranges

• Sin, cos, sec and cosec have period $2\pi$. Tan and cot have period $\pi$.
• Tan and sec are not defined at the odd multiples of $\dfrac{\pi}{2}$. Cot and cosec are not defined at the multiples of $\pi$.
• Sin and cos take the values in $[-1, 1]$. Tan and cot take every real value.
• Sec and cosec take the values in $(-\infty, -1] \cup [1, \infty)$.

Graphs

• $y = \sin x$ is a wave through the origin. $y = \cos x$ is the same wave moved $\dfrac{\pi}{2}$ to the left.
• $y = \tan x$ rises between asymptotes at the odd multiples of $\dfrac{\pi}{2}$.
• For $y = a\sin(bx + c) + d$ with $b > 0$:
  • the amplitude is $|a|$;
  • the period is $\dfrac{2\pi}{b}$;
  • the shift is $-\dfrac{c}{b}$ sideways;
  • the range is $[d - |a|, d + |a|]$.

A worked example

Find $\sec \dfrac{23\pi}{6}$.

Step 1. Whole turns. $\dfrac{23\pi}{6} = 2\pi + \dfrac{11\pi}{6}$.

Step 2. The quadrant. $\dfrac{11\pi}{6}$ is in quadrant IV, where cos and sec are positive. The acute angle is $\dfrac{\pi}{6}$.

Step 3. The value. $\cos \dfrac{11\pi}{6} = \dfrac{\sqrt{3}}{2}$, so $\sec \dfrac{23\pi}{6} = \dfrac{2}{\sqrt{3}}$.

Practice with new numbers

A program to try

The program reads three waves from their numbers and prints the amplitude, the period and the range of each.

Mistakes to avoid

• Reading $\sin^2 x$ as $\sin(x^2)$. It is $(\sin x)^2$.
• Skipping the sign. The size comes from the acute angle, and the sign from the quadrant.
• The period of $\sin bx$ as $2\pi b$. It is $\dfrac{2\pi}{b}$.
• Dividing by zero. $\tan \dfrac{\pi}{2}$ and $\operatorname{cosec} \pi$ are not defined.
• Degrees in Python. math.sin(30) is the sine of 30 radians. Write math.sin(math.radians(30)) for $30^\circ$.

Check yourself

1. In which quadrant is $x$ when $\cos x < 0$ and $\cot x < 0$?

2. Find $\sin\left(-\dfrac{7\pi}{3}\right)$.

3. What are the amplitude and the period of $y = -3\cos \dfrac{x}{4}$?

4. Why does $\sec x$ never take a value between $-1$ and $1$?

5. Which is larger, $\sin 2$ or $\sin 3$? The angles are in radians.