Defining sine and cosine —using a Unit Circle

an interactive explainer

If we take a right-angled triangle ABC where angle B is 90 degrees, the sine of an angle is defined as the ratio between the opposite side and the hypotenuse.

That is, $sin\> \angle A = \frac{opposite}{hypotenuse} = \frac{BC}{AB}$

Similarly, $cos\> \angle A = \frac{adjacent}{hypotenuse} = \frac{AC}{AB}$

A tool often used to introduce trigonometric ratios is the unit circle. Imagine a circle of radius 1 unit centered at the origin (0, 0). A right angled triangle can be drawn with these three points:

• The center of the circle, (0, 0)
• Any point on the circumference of the circle, say (x, y)
• A point on the X-axis, ie, (0, y)

Using these three points, a right angled triangle can be drawn with any (x, y) on the circumference of the circle! Try it yourself below!

A right-angled triangle drawn this way has the unique characteristic that it's hypotenuse is always of length 1 unit. So the sine and cosine can be simplified in this instance to be exactly equal to the length of the opposite side and the length of the adjacent side respectively!

Several conclusions can be drawn from the unit circle, for instance,

• The value of sin A and cos A for any angle A lies between -1 and 1.
• Sine is negative in the third and fourth quadrant, and positive in the first and second quadrant.
• Cosine is negative in the second and third quadrant, and positive in the first and fourth.