# the sine/cosine rule?

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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

Basically, every angle has a “sine” and a “cosine”, which are values unique to every angle between 0 degrees and 360 degrees.

Now assume you have a triangle. It has three sides and three angles. The side “opposite” an angle is the side not touching the angle, and vice versa.

The Law of Sines states that the ratio between the sine of an angle in a triangle and the side length of the opposite side is equal for all three angle/opposite-side pair, although this ratio changes between triangles.

The Law of Cosines states that the square of the length of a triangle’s side is equal to the sum of the squares of the other two sides, minus twice the product of the other two sides’ lengths and the cosine of the opposite angle.

These Laws let us find all three angles and side lengths of a triangle given adequate information (two angles and a side, two sides and the angle between then, or three sides), which are why they are so useful.

If this isn’t clear, I’ll post a diagram when I get home.

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.

I assume you know how sine & cosine are used (at least in terms of SOHCAHTOA) and just need help understanding these 2 rules. If you want to know what sine & cosine actually are, that’s another topic.

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**Cosine rule**
You know the Pythagorean Theorem, correct? Well, that only works for a right (90°) triangle. If it isn’t a right triangle, that’s where the cosine rule can come into play. [**Here is the proof**](https://mathspace-production-media.mathspace.co/media/imagecontent/cosine_law_proof.gif). You basically split it into 2 triangles, both now right-angled, so you use Pythagorean Theorem on both, combine them, and you get the cosine rule.

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**Sine rule**

The sine rule is just basic algebraic manipulation. [**Here is the proof**](https://www.geogebra.org/resource/ADJFzYAr/nvqaSonSqFBFKzo6/material-ADJFzYAr.png). It involves converting a triangle into 2 right angled ones like for the cosine rule, then use the SOHCAHTOA to define an angle on each triangle, isolating the 90° side, then setting the equations equal to each other.

Imagine you have a line-segment that is one length long.

One end is fixed to a horizontal line AND the angle between the horizontal and the segment, we’ll call Alpha.

Sin(Alpha) x (Length of Segment) = the height of a second line-segment that is attached to the original line segment and which drops down (vertically) to the horizontal line.

Cos(Alpha) is similar but is the distance along the horizontal line from the point where the original line-segment connects, to the point where the Sin vertical line-segment meets the horizontal line.

When original line-segment length changes, the Sin(Alpha) & Cos(Alpha) need to be products (i.e. multiplied) to get the height or length of the Sin Cos functions.

There is a really useful thing with triangles. If the angles are the same, the lengths of the sides are in proportion. This can be used in the real world to work out things like the height of a tree or a mountain.

So it comes in handy to have a shortcut of all possible triangles, because you can then use what information you have to figure out the rest. For example, you have the distance between yourself and the base of the tree, and the angle looking from the ground to the top of the tree. You can use a reference triangle with the same angle to work out the height of the tree.

To make a set of all the possible reference triangles, a Unit Circle works well. This is a circle with radius one, drawn on a Cartesian plane, which has a horizontal axis and a vertical axis. To make a unit circle, imagine an arrow drawn on a blank Cartesian plane. It starts completely flat (lying on the horizontal axis). The arrow starts at zero and the tip is exactly at one. If you keep the origin of the arrow at zero, and slowly rotate the arrow anti clockwise, the angle between the arrow and the horizontal axis will increase and over a full rotation will draw a circle with radius one.

At each step of the rotation, you can make a triangle, showing the width (along the horizontal axis) and height (along the vertical axis). You can use Pythagoras’s theorem to work out these numbers. The height is called Sine and the width Cosine.

Imagining the arrow lying on the horizontal axis. This is an angle of zero. The height, or Sine, is zero. The Cosine, one. As the arrow spins anticlockwise, the height goes from zero to a maximum of one, before decreasing back to zero, then decreasing to negative one, then up to zero as the arrow swings right around a full rotation. At the same time for the width, or cosine, starts at one, goes to zero, then foes to negative one, then back to zero, then back to one. Since the arrow can keep spinning around forever (like a backwards clock), the angle can go on forever, which is why the graph of sine and cosine look like waves that go up and down forever.