Step by step
The question
Question: find tan θ, when sin θ = 15/17.
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 θ.
2 Use what sin θ means
The opposite side is across from θ.
We write opp, adj and hyp for short.
The adjacent side is next to θ.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
sin θ is the opposite side over the hypotenuse.
The question says sin θ = 15/17.
So opp over hyp is 15/17.
Every right triangle with the angle θ has the same shape.
A bigger one could have 30 and 34.
30/34 is 15/17 too.
So we can use the small one: opp = 15 and hyp = 17.
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 = 15, adj = x and hyp = 17.
15² means 15 times 15.
That is 225.
17² means 17 times 17.
That is 289.
Write the squares: 225 + x² = 289.
We want x² by itself.
The 225 is in the way.
To remove 225, we take away 225.
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.
225 and − 225 make zero.
Cross them out.
289 minus 225 is 64.
x² is 64.
So x times x is 64.
That means x is the square root of 64.
The square root of 64 is the plus number that times itself makes 64.
8 times 8 is 64.
−8 times −8 is 64 too, but the √ sign always means the plus one.
So x = 8.
4 Write tan θ
tan θ is the opposite side over the adjacent side.
Put in opp = 15 and adj = 8.
So tan θ = 15/8.
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The sin⁻¹ button turns sin θ back into the angle: θ is about 61.9°.
tan 61.9° is about 1.87.
And 15/8 is about 1.88.
Very close: the small gap is only from rounding the angle.
So tan θ = 15/8 is right!
Let's look back.
First: draw the triangle.
Then: use what sin θ means.
Then: find the missing side.
Then: write tan θ.
Last: check.