Step by step
The question
Question: find tan A, when sin A = 5/13.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw the triangle
sin, cos and tan belong to an angle of a right triangle.
So we draw a right triangle with the angle A.
2 Use what sin A means
The opposite side is across from A.
We write opp, adj and hyp for short.
The adjacent side is next to A.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
sin A is the opposite side over the hypotenuse.
The question says sin A = 5/13.
So opp over hyp is 5/13.
Every right triangle with the angle A has the same shape.
A bigger one could have 10 and 26.
10/26 is 5/13 too.
So we can use the small one: opp = 5 and hyp = 13.
3 Find the missing side
The adjacent side is missing.
We call it x.
Pythagoras: in a right triangle, the two short sides squared add up to the hypotenuse squared.
Now put the numbers in: opp = 5, adj = x and hyp = 13.
5² means 5 times 5.
That is 25.
13² means 13 times 13.
That is 169.
Write the squares: 25 + x² = 169.
Is x alone now?
Not yet: + 25 is still with x.
How do we remove it?
We want x² by itself.
The 25 is in the way.
To remove 25, we take away 25.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
25 and − 25 cancel each other.
They make zero.
169 minus 25 is 144.
x² is 144.
So x times x is 144.
That means x is the square root of 144.
The square root of 144 is the plus number that times itself makes 144.
12 times 12 is 144.
−12 times −12 is 144 too, but the √ sign always means the plus one.
So x = 12.
So after step 3: we took away 25, and now we have x = 12.
4 Write tan A
tan A is the opposite side over the adjacent side.
Put in opp = 5 and adj = 12.
So tan A = 5/12.
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
The right way: tan A = 5/12.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The sin⁻¹ button turns sin A back into the angle: A is about 22.6°.
tan 22.6° is about 0.42.
And 5/12 is about 0.42.
The same.
So tan A = 5/12 is right!
6 Pick the choice
Our answer is tan A = 5/12.
That is choice A.
7 Why not the others
B is 12/13.
Wrong: that is cos A: adjacent over hypotenuse.
C is 13/12.
Wrong: hypotenuse over adjacent: that is not tan.
D is 12/5.
Wrong: upside down: tan is opposite over adjacent.
Let's look back.
First: draw the triangle.
Then: use what sin A means.
Then: find the missing side.
Then: write tan A.
Then: check.
Then: pick the choice.
Last: why not the others.