Step by step
The question
Question: find tan B.
Triangle ABC has a right angle at C, and tan A = 5/2.
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 tan 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.
tan A is the opposite side over the adjacent side.
The question says tan A = 5/2.
So opp over adj is 5/2.
Every right triangle with the angle A has the same shape.
A bigger one could have 10 and 4.
10/4 is 5/2 too.
So we can use the small one: opp = 5 and adj = 2.
3 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: 2.
The side next to B is the upright side: 5.
Put in opp = 2 and adj = 5.
So tan B = 2/5.
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
4 Check
Let's check with the calculator.
Set the calculator to degrees first.
The tan⁻¹ button turns tan A back into the angle: A is about 68.2°.
The two small angles share 90°.
So B is about 21.8°.
tan 21.8° is about 0.4.
And 2/5 is 0.4.
The same.
So tan B = 2/5 is right!
Let's look back.
First: draw the triangle.
Then: use what tan A means.
Then: write tan B.
Last: check.