Step by step
The question
Question: find cos A, when tan A = 9/40.
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 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 = 9/40.
So opp over adj is 9/40.
Every right triangle with the angle A has the same shape.
A bigger one could have 18 and 80.
18/80 is 9/40 too.
So we can use the small one: opp = 9 and adj = 40.
3 Find the missing side
The hypotenuse 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 = 9, adj = 40 and hyp = x.
9² means 9 times 9.
That is 81.
40² means 40 times 40.
That is 1600.
Write the squares: 81 + 1600 = x².
81 plus 1600 is 1681.
x² is 1681.
So x times x is 1681.
That means x is the square root of 1681.
The square root of 1681 is the plus number that times itself makes 1681.
41 times 41 is 1681.
−41 times −41 is 1681 too, but the √ sign always means the plus one.
So x = 41.
Careful!
We cannot just add the sides.
Pythagoras adds the squares: 81 + 1600 = 1681.
Then the square root.
4 Write cos A
cos A is the adjacent side over the hypotenuse.
Put in adj = 40 and hyp = 41.
So cos A = 40/41.
5 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 12.7°.
cos 12.7° is about 0.98.
And 40/41 is about 0.98.
The same.
So cos A = 40/41 is right!
Let's look back.
First: draw the triangle.
Then: use what tan A means.
Then: find the missing side.
Then: write cos A.
Last: check.