Step by step
The question
Question: find x.
cos 45° = sin x°.
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 45° 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 45°.
The other one gets the rest: 90 minus 45.
90 minus 45 is 45.
So the other angle is 45°.
3 Name the sides from each angle
Look at side b.
From the 45° angle, it is next to it: the adjacent side.
We write opp, adj and hyp for short.
From the top angle, the same side b is across: the opposite side.
Side c is the hypotenuse for both angles.
It is across from the right angle.
cos is the adjacent side over the hypotenuse.
From 45°: b over c.
sin is the opposite side over the hypotenuse.
From the top angle: b/c too.
Same two sides, same fraction.
So cos 45° = sin 45°.
This works in every right triangle.
The cos of one small angle is the sin of the other.
Together they make 90°.
4 Find x
The question says cos 45° = sin x°.
We found cos 45° = sin 45°.
So sin x° = sin 45°.
Between 0° and 90°, each angle has its own sin.
So x = 45.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
cos 45° is about 0.7071.
sin 45° is about 0.7071 too.
They are the same.
So x = 45 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.