Step by step
The question
Question: find sin A, when tan A = 1.
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 = 1.
1 is the same as 1/1.
So opp over adj is 1/1.
Every right triangle with the angle A has the same shape.
A bigger one could have 2 and 2.
2/2 is 1/1 too.
So we can use the small one: opp = 1 and adj = 1.
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 = 1, adj = 1 and hyp = x.
1² means 1 times 1.
That is 1.
1² means 1 times 1.
That is 1.
Write the squares: 1 + 1 = x².
1 plus 1 is 2.
x² is 2.
So x times x is 2.
That means x is the square root of 2.
2 is not a square number.
Its root is a decimal that never ends.
So we keep √2.
That is the exact length.
Careful!
We cannot just add the sides.
Pythagoras adds the squares: 1 + 1 = 2.
Then the square root.
4 Write sin A
sin A is the opposite side over the hypotenuse.
Put in opp = 1 and hyp = √2.
Books do not leave a square root at the bottom.
We multiply by √2 over √2.
That is 1, so the value stays the same.
√2 times √2 is 2: that is what a square root means.
On top: 1 times √2 is √2.
On the bottom: √2 times √2 is 2.
So sin A = √2/2.
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 45.0°.
sin 45.0° is about 0.71.
And √2/2 is about 0.71.
The same.
So sin A = √2/2 is right!
Let's look back.
First: draw the triangle.
Then: use what tan A means.
Then: find the missing side.
Then: write sin A.
Last: check.