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Suppose ABC is a right triangle with sides of lengths a b and c and the right angle at C find the unknown side length using the Pythagorean Theorem, and then find the values of the six trig function for angle B.

11) a= 5 b= 12

c= 13 so how do I find the angles? I know I have 90 degree + a + b = 180 but that is it.

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    In a right triangle, the values for the six standard trig functions of an angle can be computed as the ratios of certain sides.2011-06-05

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In general, when we talk about a triangle and we've named the vertices with capital letters (your $\triangle ABC$ has vertices $A$, $B$, and $C$), we name the lengths of the sides opposite each vertex with the lower-case version of the letter labeling that vertex (the side opposite $B$ in your triangle is called $b$, which is the side with endpoints $A$ and $C$).

Since you've talked about $x$, $y$, and $r$ in other comments, I think you may be working with trigonometry on a coordinate plane. Let's try drawing a picture of the situation. We're talking about finding a trig function of angle $B$, so we want to put $B$ at the origin. We want the right angle, which is at $C$ to be on the $x$-axis. Since $a=5$ is the length of the side between $B$ and $C$, let's put $C$ at $5$ on the $x$-axis. With a right angle at $C$, point $A$ will be directly above or below $C$. Since $b=12$ is the length of the side between $A$ and $C$, let's put $A$ at $(5,12)$, directly above $C$.

Here's the picture so far:

diagram as described above

Given this picture, do you know what the $x$, $y$, and $r$ are for finding the trig functions of angle $B$?

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    @Adam: That's right. Now, if you have $\triangle DEF$ with right angle at $D$, $e=7$, and $f=24$, what is $\sin F$?2011-06-05
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Use the Law of Cosines. (Technically, you could also use the Law of Sines, but given your starting data the Law of Cosines would be easier to use).

Edit: Or, if you didn't want to use dynamite when a pick-axe would be better, just use the basic trig functions and solve for the ratio of the sides.

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    That is correct, so how do I figure out what the angle of$b$is?2011-06-05
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In a right triangle, $\sin$ and $\cos$ are paricularly easy to calculate, since in this case $\sin B$ is just the ratio of the opposite over the hypotenuse, and $\cos B$ is the ratio of the adjacent over the hypotenuse. To find $\tan B$, $\cot B$, $\csc B$, $\sec B$, just express them in terms of $\sin B$ and $\cos B$ and simplify.

I assume $B$ is the angle opposite $b$, so if you draw your triangle out, you should see $\sin B=\frac{12}{13}$. The rest follows quite like this.

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    Kind of yes, the triangle can be flipped and turned around but the legs and the angles are all the same relative to each other. Since all the legs and angles move around in the same way, the values will always be the same, so it doesn't matter how you label really.2011-06-05