Difference Between Trigonometric Sine And Cosine

In trigonometric studies, we always come across 2 main functions, sine and cosine.

(Another function is Tangent, which is the division of sine by cosine).

Their shapes or profiles are similar besides having same amplitude.

So why the need to have these 2 main trigonometric functions when they are similar?

Though they looked the same in term of their shapes or timing profiles, they do possess one key difference.

A simple explanation can be done with some numbers (angles) filled into the trigonometric functions.

At the origin (angle A = 0⁰),
sin 0 = 0
cos 0 = 1 (peak value)

At angle = 90⁰,
sin 90 = 1 (peak value)
cos 90 = 0

At angle = 180⁰,
sin 180 = 0
cos 180 = -1 (reverse peak)

At angle = 270⁰,
sin 270 = -1 (reverse peak)
cos 270 = 0

From the pattern of rotational values, you can see that at angle = 0⁰, cosine has a value of 1, which sine will have, only at angle = 90⁰. There is a delay of 90⁰ !

Likewise for the value of "-1", there is a delay of 90⁰ between cosine and sine.

Mathematically, you can then write the relationship between Sine and Cosine as

sin A = cos (A - 90).

The sine function is said to "lag" the cosine function by 90⁰.

Therefore, due to the 90 degree difference, we need to have both Sine and Cosine existing, since trigonometric function has both amplitude and angle information embedded within them. Though their shapes are the same, their angle positioning are different.

This knowledge is useful as it helps to form many types of waveforms and also helps in the analysis of speech and communication signal, and in the studies of noise reduction.

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