Ok, so I am a freshman in high school and I want to know how the sine, cosine, and tangent help you find the lengths of sides. (This is not a homework question)

I know that in a 30-60-90 degree right triangle, the relationship is 1, square root of 3, and the hypotenuse is two.

I also understand the following:

  1. Sine is Opposite over Hypotenuse
  2. Cosine is Adjacent over Hypotenuse
  3. Tangent is Opposite over Adjacent

The main thing I have a problem with is determining which sides are what; which one is opposite and which one is adjacent.

For instance, which angle would you use for sin30 (where angle c is the right angle, a is the top angle, and b is the angle to the right) and which angle would you use for sin60?

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    $\begingroup$ opposite and adjacent sides are with respect to an angle. when you switch the angle these sides switch too. $\endgroup$ – abel Feb 13 '15 at 17:12
  • $\begingroup$ Give examples please and elaborate; ty @abel $\endgroup$ – Isaiah Feb 13 '15 at 17:13
  • $\begingroup$ Also, feel free to reformat my post; I don't know latex formatting $\endgroup$ – Isaiah Feb 13 '15 at 17:26

i can't draw a picture here that would make it easier to see what i am saying. label the right angle triangle $ABC$ so that $\angle C = 90^\circ.$

let us concentrate on the $\angle A.$ the opposite side is $BC$ and the adjacent side is $AC.$

if you switch to the the $\angle B,$ then the opposite side is $AC$ and the adjacent side is $BC.$

does this make sense?

  • $\begingroup$ Ok, how do you determine which angle to examine. Sorry if I sound silly, my regular math class hasn't gotten this far yet. $\endgroup$ – Isaiah Feb 13 '15 at 17:19
  • $\begingroup$ that depends on the data in the problem. if you have a concrete problem you are working on we can discuss that. doing mathematics in abstract is hard; it gets easier when you have a particular problem to solve. $\endgroup$ – abel Feb 13 '15 at 17:20
  • $\begingroup$ Ok, as an example, show me the angle we would concentrate on for sin30 then sin60 $\endgroup$ – Isaiah Feb 13 '15 at 17:22
  • $\begingroup$ You need to consider an equilateral triangle length 2 for the sides. Chop it into 2 with a perpendicular line from the base. $\endgroup$ – Karl Feb 13 '15 at 17:24
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    $\begingroup$ we will look at the angle $30^\circ$ and the $1, \sqrt3, 2$ triangle. the opposite side has length $1$ and adjacent has length $\sqrt 3$. so the trig ration $\sin 30^\circ = \frac{opp}{hyp} = 1/2$ and $\cos 30^\circ = \sqrt3/2$ $\endgroup$ – abel Feb 13 '15 at 17:26

Consider you have triangle as below. $A, B, C$ are its vertices, $a, b, c$ are its sides, $\alpha, \beta, \gamma$ are its angles (I haven't marked $\beta$ and $\gamma$, so let's describe all we need for $\alpha$).
$\alpha$ is an acute angle of right triangle $ABC$. Side $BC$ (or $a$, which is name given to it using other notation) is its opposite cathetus, while $AC$ (or $b$) is its adjacent cathetus. $AB$ (or $c$) is a hypothenuse.
Considering this, for example $\sin \alpha = \frac{\text{opposite}}{\text{hypothenuse}} = \frac{BC}{AB} = \frac ac$. It's very similar for other functions, hope you're able to manage with them.
As @abel correctly mentioned, trigonometrical functions are defined not for a triangle but for an angle, so you need to check that the triangle is right, find a given acute angle in it and then you will have two sides adjacent to this angle (hypothenuse and adjacent cathetus) and one side opposite to it.

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Hypotenuse is the longest side usually labelled H. It is opposite the right angle.

The other sides depend on which angle you are interested in.

It might help that adjacent is a fancy word for 'next to'. So the angle you are labelling the sides in relation to will be between the hypotenuse and the adjacent. It is next to the angle. Usually called A

The final angle is the one opposite the angle you chose to label the sides with. It is called O usually.

The angle you are interested in will vary from problem to problem.


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