The sine function is defined for a right triangle as
the ratio of the opposite side of an angle to the hypoteneuse.
What is the relationship between this definition
and the sine curve ?
In the Animation below we consider the motion of a partice on the unit circle.
At each time a right triangle is formed with hypoteneuse 1. Therefore the value of sin(t) is precisely the y coordinate. Simultaneously
We plot the y coordinate on a linear scale generating the basic Sine Curve.
Basic Sine Curve
We now do the same thing to generate the basic Cosine
Curve. The Cosine is defined as the ratio of the adjacent side to the hypoteneuse. This time the value of the cosine is the
x coordinate which is plotted on the vertical ont the linear graph
Basic Cosine Curve
An application - suppose we have a mass on a spring
initially compressed. It is released and oscillates up and down. As the following animation shows if we plot its position
vs time we get a cosine curve.
Trig functions are used to model any application in which there is periodic motion.
Mass on a Spring
The sines function is an odd Function i.e. sin(-t)
In the animation below we see the sine curve generated simultaneously for positive and negative values
sin(-t) = -sin(t)
The cosine function is an even function i.e. cos(-t)
cos(-t) = cos(t) animation