Home >> PURE MATHS, Trigonometry, sine, cosine, tangent

The General Angle

Consider a radius of length '1' rotating anti-clockwise about the origin.

The coordinates of any point on the circle give the values of the adjacent and opposite sides of a right angled triangle, with the radius the hypotenuse.

The General Angle ( θ theta) is the included angle between the radius and the x-coordinate of the point.

As the radius rotates the x and y values change. Hence the values of sine, cosine and tangent also change.

The result is summarized in the diagram below.

Example #1

Example #2

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Sine

Points of interest :

The 'sine graph' starts at zero.

It repeats itself every 360 degrees(or 2 pi).

y is never more than 1 or less than -1

(vertical displacement from the x-axis is called the amplitude).

A 'sin graph' leads a cos graph by 90 degrees

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Cosine

Points of interest:

The 'cosine graph' starts at one.

It repeats itself every 360 degrees(or 2 pi).

y is never more than 1 or less than -1

(displacement from the x-axis is called the amplitude).

A 'cos graph' lags a 'sin graph' by 90 degrees(pi/2) - this is termed a phase shift

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Tangent

Points of interest:

The 'tangent graph' starts at zero.

It repeats itself every 180 degrees.

y can vary between numbers approaching infinity and minus infinity.

Further Comparison

Only the cosine function is symmetrical (a mirror image of itself) about the y-axis.

All the functions are cyclic - a waveform along the horizontal axis is repeated.

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