Step by step
The question
Question: find tan B.
Triangle ABC has a right angle at C, and cos A = 9/41.
1 Draw the triangle
Triangle ABC has its right angle at C.
So A and B are the two small angles.
We draw it with A at the bottom.
2 Use what cos 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.
cos A is the adjacent side over the hypotenuse.
The question says cos A = 9/41.
So adj over hyp is 9/41.
Every right triangle with the angle A has the same shape.
A bigger one could have 18 and 82.
18/82 is 9/41 too.
So we can use the small one: adj = 9 and hyp = 41.
3 Find the missing side
The opposite 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 = x, adj = 9 and hyp = 41.
9² means 9 times 9.
That is 81.
41² means 41 times 41.
That is 1681.
Write the squares: x² + 81 = 1681.
We want x² by itself.
The 81 is in the way.
To remove 81, we take away 81.
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.
81 and − 81 make zero.
Cross them out.
1681 minus 81 is 1600.
x² is 1600.
So x times x is 1600.
That means x is the square root of 1600.
The square root of 1600 is the plus number that times itself makes 1600.
40 times 40 is 1600.
−40 times −40 is 1600 too, but the √ sign always means the plus one.
So x = 40.
4 Write tan B
tan B is the opposite side over the adjacent side.
But the picture names the sides from A.
Now we look from B, the top angle.
The side across from B is the bottom side: 9.
The side next to B is the upright side: 40.
Put in opp = 9 and adj = 40.
So tan B = 9/40.
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The cos⁻¹ button turns cos A back into the angle: A is about 77.3°.
The two small angles share 90°.
So B is about 12.7°.
tan 12.7° is about 0.23.
And 9/40 is about 0.23.
The same.
So tan B = 9/40 is right!
Let's look back.
First: draw the triangle.
Then: use what cos A means.
Then: find the missing side.
Then: write tan B.
Last: check.