the sine/cosine rule?

Question

I am autistic and have difficulty understanding certain things for (sometimes) no apparent reason. This is one of those times. I am in need of help, and would appreciate some explanation. Thankyou! Please simplify it as much as possible.

In: 7

Answer 1

Not a direct answer, but hopefully will give some context to the other replies.

Sine and cosine express the relationship between an angle and the amount of linear displacement from an axis the angle produces at the point where it crosses the outside of a circle of radius = 1. Sine refers to vertical displacement, and cosine to horizontal displacement.

Draw an x/y axis and put a circle in the middle with a radius of 1 (unit doesn’t matter).

Now from the origin (the centre), draw a line so it crosses the circle’s circumference. Measure the angle between the line and the x axis. Also measure vertically from where the line crosses the circumference to the x axis. The ratio between the length of this vertical distance to the circle’s radius is the angle’s sine.

Cosine is the same. Except while the line’s angle is still measured from the x axis, we take the intercept’s horizontal distance from the y axis, instead of it’s vertical distance to the x axis.

So the sine of a 30 degree angle is 0.5, because a 30 degree line touches the circle at a displacement from the x axis of half the circle’s radius. The sine of 60 degrees is ~0.87 because the vertical distance from where a 60 degree angle intercepts the circle is ~.87 of its radius. The sine of 90 degrees is 1 because the displacement and the radius are the same distance. Likewise, the sine of 0 degrees is 0.

As the angle increases past 90 degrees and beyond, the sine corresponds to the ratio for distances above and below the x axis as before. Thus the sine of 120 degrees is also ~0.87, 150 degrees is 0.5, and 180 degrees is 0. Between 180 degrees and 360 degrees, sines are negative because the intercept is below the x axis.

Sines repeat in multiples of 360 degrees. Cosines are always 90 degrees from their sines, hence the name co-sine.

[This website](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html) has a fun interactive tool to help gain an intuitive appreciation for how sine, cosine, and tangent relate to each other.