Trigonometry is composed of two words: Trigon and Metron. Trigon means ‘triangle’ and metron means ‘to measure’. Combined, it means measuring sides or angles of a triangle and that is what trigonometry precisely is. It is the study of angles of triangles and relationships between them. Trigonometric Functions can be defined as ‘The functions which help us relate the angles and sides of triangles.’
Trigonometric functions are also known as circular functions. This is because they are normally explained and derived using unit circles (circles with radius equal to 1 unit). Let us define these functions now.
Trigonometric Ratios:
In earlier grades, we have seen the trigonometric ratios for acute angles (angles less than \(90^{\circ}\)). But those ratios are also defined for obtuse angles, reflex angles, negative angles, etc. Let us recall the concepts that we have already seen before.
In figure 1, \(\triangle ABC\) is a rectangular or right angled triangle. If we take \(\angle ACB\) = \(\theta\),
\(\sin~\theta\) = \(\large\frac{AB}{AC}\) = \(\large\frac{Opposite Side}{Hypotenuse}\) = \(\large \frac{1}{\csc~\theta}\)
\(\cos~\theta\) = \(\large\frac{BC}{AC}\) = \(\large\frac{Adjacent Side}{Hypotenuse}\) = \(\large \frac{1}{\sec~\theta}\)
\(\tan~\theta\) = \(\large \frac{\sin~\theta}{\cos~\theta}\) = \(\large\frac{AB}{BC}\) = \(\large \frac{Opposite Side}{Adjacent Side}\) = \(\large\frac{1}{\cot~\theta}\)
Let us take a unit circle as shown in the figure 2. The radius of the circle is 1 unit. The length of the arc AP is \(x\) units. So, the angle subtended at the centre which is origin in this case, is \(x\) radians. If we take any random point on the circle (\(P\)), its coordinates are \((m,n)\). Using the knowledge of the ratios, we can define them as follows:
- \(sin~x\) = \(\large\frac{PM}{OP}\) = \(\large \frac{n}{1}\) = \(n\)
- \(cos~x\) = \(\large \frac{OM}{OP}\) = \(\large\frac{m}{1}\) = \(m\)
\(\triangle OMP\) is a right angled triangle. So, applying Pythagoras theorem, we get:
\(OM^2 + MP^2\) = \(OP^2\)
\(m^2 + n^2\) = \(1^2\)
From the results we obtained before, we get
\(\cos^2~x + \sin^2~x\) = \(1\)
Dividing both the sides by \(\cos^2~x\), we get
\(1 + \tan^2~x\) = \(\sec^2~x\)
Dividing both sides now by \(tan^2~x\), we get
\(1 + cot^2~x\) = \(\csc^2~x\)
The identities written above are true for any point taken on the circle. The circumference of the circle is equal to 2π units. Also, the full angle is equal to 2π radians. So,
\(∠AOB\) = \(90°\) = \(\large \frac{π}{2}\) radians
\(∠AOD\) = \(270°\) = \(\large \frac{3π}{2}\) radians
If a given angle is an integral multiple of \(\large \frac{π}{2}\), it is known as quadrantal angle. Let us move along the circle starting from P. After 1 full revolution, we reach back at P. Isn’t this movement periodic?
Yes, it is. This is the reason due to which these functions are said to be periodic. Even after n revolutions, at P, value of sin x will still be equal to m. This means
\(\sin~(nπ+x)\) = \(\sin~x\)
There are similar conclusions that can be drawn for other ratios as well. This also means that the equation \(\sin~x\) = \(0\) will have infinite roots! Generally, the roots are given by
\(x\) = \(nπ\), where n is any integer
Similarly, if \(\cos~x\) = \(0\),
\(x\) = \((2n+1)\tfrac{\pi}{2}\), where n is any integer.
Table 1: Values of different trigonometric ratios
The following table gives the value of all the trigonometric ratios for common angles.
Angles (In Degrees) |
0° |
30° |
45° |
60° |
90° |
180° |
270° |
360° |
Angles (In Radians) |
0 |
\(\frac{π}{6}\) |
\(\frac{π}{4}\) |
\(\frac{π}{3}\) |
\(\frac{π}{2}\) |
π |
\(\frac{3π}{2}\) |
2π |
sin |
0 |
\(\frac{1}{2}\) |
\(\frac{1}{√2}\) |
\(\frac{√3}{2}\) |
1 |
0 |
-1 |
0 |
cos |
1 |
\(\frac{√3}{2}\) |
\(\frac{1}{√2}\) |
\(\frac{1}{2}\) |
0 |
-1 |
0 |
1 |
tan |
0 |
\(\frac{1}{√3}\) |
1 |
√3 |
N D |
0 |
N D |
0 |
cot |
N D |
√3 |
1 |
\(\frac{1}{√3}\) |
0 |
N D |
0 |
N D |
csc |
N D |
2 |
√2 |
\(\frac{2}{√3}\) |
1 |
N D |
-1 |
N D |
sec |
1 |
\(\frac{2}{√3}\) |
√2 |
2 |
N D |
-1 |
N D |
1 |
To learn how to decide signs of trigonometric functions in different quadrants and how to calculate their value, download BYJU’S-The Learning App and experience the fun in learning.