Step by step
The question
Question: find cos A, when sin A = 1/3.
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 = 1/3.
So opp over hyp is 1/3.
Every right triangle with the angle A has the same shape.
A bigger one could have 2 and 6.
2/6 is 1/3 too.
So we can use the small one: opp = 1 and hyp = 3.
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 = 1, adj = x and hyp = 3.
1² means 1 times 1.
That is 1.
3² means 3 times 3.
That is 9.
Write the squares: 1 + x² = 9.
We want x² by itself.
The 1 is in the way.
To remove 1, we take away 1.
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.
1 and − 1 make zero.
Cross them out.
9 minus 1 is 8.
x² is 8.
So x times x is 8.
That means x is the square root of 8.
8 is not a square number.
But it has a square number in it: 4.
8 is 4 times 2.
So x is the square root of 4 times 2.
A root over a times can be split into two roots.
So it is √4 times √2.
√4 is 2, because 2 times 2 is 4.
We leave out the times sign.
So x = 2√2.
That is the exact length.
Careful!
We cannot just take away the sides.
Pythagoras works with the squares: 9 − 1 = 8.
Then the square root.
4 Write cos A
cos A is the adjacent side over the hypotenuse.
Put in adj = 2√2 and hyp = 3.
So cos A = 2√2/3.
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 19.5°.
cos 19.5° is about 0.94.
And 2√2/3 is about 0.94.
The same.
So cos A = 2√2/3 is right!
Let's look back.
First: draw the triangle.
Then: use what sin A means.
Then: find the missing side.
Then: write cos A.
Last: check.