Step by step
The question
Question: find sin E.
Triangle DEF has a right angle at F, and tan D = 7/24.
1 Draw the triangle
Triangle DEF has its right angle at F.
So D and E are the two small angles.
We draw it with D at the bottom.
2 Use what tan D means
The opposite side is across from D.
We write opp, adj and hyp for short.
The adjacent side is next to D.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
tan D is the opposite side over the adjacent side.
The question says tan D = 7/24.
So opp over adj is 7/24.
Every right triangle with the angle D has the same shape.
A bigger one could have 14 and 48.
14/48 is 7/24 too.
So we can use the small one: opp = 7 and adj = 24.
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 = 7, adj = 24 and hyp = x.
7² means 7 times 7.
That is 49.
24² means 24 times 24.
That is 576.
Write the squares: 49 + 576 = x².
49 plus 576 is 625.
x² is 625.
So x times x is 625.
That means x is the square root of 625.
The square root of 625 is the plus number that times itself makes 625.
25 times 25 is 625.
−25 times −25 is 625 too, but the √ sign always means the plus one.
So x = 25.
Careful!
We cannot just add the sides.
Pythagoras adds the squares: 49 + 576 = 625.
Then the square root.
4 Write sin E
sin E is the opposite side over the hypotenuse.
But the picture names the sides from D.
Now we look from E, the top angle.
The side across from E is the bottom side: 24.
The side next to E is the upright side: 7.
The hypotenuse stays 25.
Put in opp = 24 and hyp = 25.
So sin E = 24/25.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The tan⁻¹ button turns tan D back into the angle: D is about 16.3°.
The two small angles share 90°.
So E is about 73.7°.
sin 73.7° is about 0.96.
And 24/25 is 0.96.
The same.
So sin E = 24/25 is right!
Let's look back.
First: draw the triangle.
Then: use what tan D means.
Then: find the missing side.
Then: write sin E.
Last: check.