Mathematics Center - Classrooms - Introduction to Trigonometric Functions

Introduction to Trigonometric Functions

What are trigonometric functions? Let’s get started from the most basic principle.

1.     The Basic Idea
The sketch is an introduction to the idea behind trigonometric functions.

Press "color" button to change the color of the gondolas.
Observe the motion of the red gondola.
Press "left" or "right" button several times to change the viewing angle.
Key question: How does the height of the red gondola changes?
Answer: >> Sine of the angle rotated<<

                     (Mouse select to view)


This sketch gives a impression of sine.
The length of the rope which is hanging down from the crane is sin t when the length of crane = 1 and angle = t.

What is length of the rope, when the length of crane = 2, 3, 4,･･･ ?.



2.     Introduction to Trigonometric Functions
Sine Function

The figure shows a definition of sine function.

The unit circle is the circle with its center at the origin and a radius of 1. Angle x is formed by rotating OA about the origin to OP. Then the y-coordinate of point P is sin x.


Cosine Function

The figure shows a definition of cosine function.

The unit circle is the circle with its center at the origin and a radius of 1. Angle x is formed by rotating OA about the origin to OP. Then the x-coordinate of point P is cos x.


Tangent Function

The figure shows a definition of tangent function.

The unit circle is the circle with its center at the origin and a radius of 1. Angle x is formed by rotating OA about the origin to OP. Point Q is the intersection of line OP and x=1. Then the y-coordinate of point Q is tan x.


Six Trigonometric Functions

Here is a combined picture of the six trigonometric functions. They are called sine, cosine, secant, cosecant, tangent and cotangent respectively.


Sine Function Plot

Press "Draw" button, or "+" button to generate the graph of the sine function.


Cosine Function Plot

Press "Draw" button, or "+" button to generate the graph of the sine function.


Tangent Function Plot

Press "Draw" button, or "+" button to generate the graph of the sine function.


3.     Graph of More Trigonometric Functions
The Graph of y = sin ax

Press "Draw" button, or "+" button to generate the graph.

The red curve is y=sin ax and the blue is y=sin x. The radius for y=sin ax rotates 'a' times faster than the radius for y=sin x. Change 'a' using the scroll bar, then find the relation of the wave length and the value of 'a'.


The graph of y=a sin b(x-c)
(Transformation of Graphs)

The sketch shows the graph of y=a sin b(x-c).

Change the value of 'a', 'b', and 'c' and observe the graph.

"Transform" button shows the following transformation process.

       y=sinx
→ y=a sin x
→   y=a sin bx
→   y=a sin b(x-c).