Step by step
The question
Question: find x.
cos x° = sin 18°.
x is between 0 and 90.
1 Draw the triangle
sin and cos come from the sides of a right triangle.
So we draw one, with the angle 18° at the bottom.
The other small angle is at the top.
We do not know it yet.
2 Find the other angle
The three angles of a triangle add up to 180°.
The right angle takes 90°.
180 minus 90 is 90.
So the two small angles share 90°.
One small angle is 18°.
The other one gets the rest: 90 minus 18.
90 minus 18 is 72.
So the other angle is 72°.
3 Name the sides from each angle
Look at side a.
From the 18° angle, it is across: the opposite side.
We write opp, adj and hyp for short.
From the top angle, the same side a is next to it: the adjacent side.
Side c is the hypotenuse for both angles.
It is across from the right angle.
sin is the opposite side over the hypotenuse.
From 18°: a over c.
cos is the adjacent side over the hypotenuse.
From the top angle: a/c too.
Same two sides, same fraction.
So sin 18° = cos 72°.
This works in every right triangle.
The sin of one small angle is the cos of the other.
Together they make 90°.
4 Find x
The question says sin 18° = cos x°.
We found sin 18° = cos 72°.
So cos x° = cos 72°.
Careful!
sin and cos of the same angle are not equal.
sin 18° is about 0.309, but cos 18° is about 0.9511.
Between 0° and 90°, each angle has its own cos.
So x = 72.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
sin 18° is about 0.309.
cos 72° is about 0.309 too.
They are the same.
So x = 72 is right!
Let's look back.
First: draw the triangle.
Then: find the other angle.
Then: name the sides from each angle.
Then: find x.
Last: check.