What is Trigonometry?
Trigonometry is that branch of Mathematics that deals with the angles and sides of triangles. Of particular interest to trigonometry is the right angled triangle, where one angle is 90. It helps to find missing angles if the magnitude of sides are known and vice versa.
A right angled triangle, in trigonometry, reveals:
A right angled triangle
• The box in the right corner indicates the right angle in the triangle.
• The known angle here is ‘x’. Many textbooks depict it commonly as (theta).
• The side opposite the right angle is the longest side and is called the hypotenuse.
• The side opposite the known angle is called the opposite.
• The third side is called the adjacent.
Three basic functions governing trigonometry are:
Estimation in terms of sides of a triangle
sin = Opposite / Hypotenuse (O/H)
cos = Adjacent / Hypotenuse (A/H)
tan = Opposite / Adjacent (O/A)
It is easy to draw from the table above, that:
tan = sin/ cos
sin = cos x tan
Thus, the relation between sin, cos and tan is verified. You may take any numerical value for an angle and using sin and cos values for the angle, verify the result with the tan value for the angle and see for yourself that the relation holds indeed!
Trigonometry in a Circle
Next, we shall aim to understand trigonometry in circles. A circle may be divided into four quadrants in the Cartesian plane which traditionally takes the center of the circle to be the origin (0,0). The conventional Cartesian rules hold here i.e. Quadrant I takes positive values of both, x and y; Quadrant II takes positive y value and negative x value; and so on for Quadrants III and IV. See coordinate geometry to learn more about Cartesian planes and coordinates.
The trigonometric circle
The radius of a circle, if rotated in anti-clockwise direction, creates different angles with the origin. Let one such angle be Thus, values of sin, cos and tan can be found for the angle ”, as shown in the above diagram. Also note that the diagram defines the trigonometric angles for a unit circle.
What is a unit circle?
Simply, a circle with radius equals one unit is called a ‘unit circle’. Values for sin, cos and tan can be easily found for a unit circle.
In graphing the sin function, we take the angle values on the horizontal axis and the corresponding sin values on the vertical axis. In varying the angles and noting the sin values for the same, you will notice that sin values vary from -1 to 1. Interestingly, sin follows a periodic graph, meaning that the graph repeats its shape after every successive period. Here, the case is 360or 2 radian.
Graph for sin and cos function
Similarly, the graph for cos function can be traced out to look like the curve as shown above.
The tan function looks slightly different as it is undefined at but crosses 0 after every 180,as shown below:
Graph for tan function
Case of non-right angled triangle
Trigonometry holds for non-right angled triangles as well, where the sin rule is:
And, the cos rule is:
+ – 2ab cos (C)