Step by step
The question
Question: find sin A, when cos A = 2/3.
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 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 = 2/3.
So adj over hyp is 2/3.
Every right triangle with the angle A has the same shape.
A bigger one could have 4 and 6.
4/6 is 2/3 too.
So we can use the small one: adj = 2 and hyp = 3.
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 = 2 and hyp = 3.
2² means 2 times 2.
That is 4.
3² means 3 times 3.
That is 9.
Write the squares: x² + 4 = 9.
We want x² by itself.
The 4 is in the way.
To remove 4, we take away 4.
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.
4 and − 4 make zero.
Cross them out.
9 minus 4 is 5.
x² is 5.
So x times x is 5.
That means x is the square root of 5.
5 is not a square number.
Its root is a decimal that never ends.
So we keep √5.
That is the exact length.
Careful!
We cannot just take away the sides.
Pythagoras works with the squares: 9 − 4 = 5.
Then the square root.
4 Write sin A
sin A is the opposite side over the hypotenuse.
Put in opp = √5 and hyp = 3.
So sin A = √5/3.
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 48.2°.
sin 48.2° is about 0.75.
And √5/3 is about 0.75.
The same.
So sin A = √5/3 is right!
Let's look back.
First: draw the triangle.
Then: use what cos A means.
Then: find the missing side.
Then: write sin A.
Last: check.