Step by step
The question
Question: find x.
sin 35° = cos(x + 15)°.
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 35° 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 35°.
The other one gets the rest: 90 minus 35.
90 minus 35 is 55.
So the other angle is 55°.
3 Name the sides from each angle
Look at side a.
From the 35° 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 35°: 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 35° = cos 55°.
This works in every right triangle.
The sin of one small angle is the cos of the other.
Together they make 90°.
4 Write the equation
The question says sin 35° = cos(x + 15)°.
We found sin 35° = cos 55°.
So cos(x + 15)° = cos 55°.
Careful!
sin and cos of the same angle are not equal.
sin 35° is about 0.5736, but cos 35° is about 0.8192.
Between 0° and 90°, each angle has its own cos.
So the angle x + 15 must be 55.
Now we solve it.
5 Remove the + 15
We want x by itself.
The + 15 is in the way.
To remove + 15, we take away 15.
We do it on both sides, so they stay equal.
+ 15 and − 15 make zero.
Cross them out.
55 − 15 = 40.
So x = 40!
6 Check
Let's check: put x = 40 in the angle.
x + 15 is 55.
Now with the calculator.
Set the calculator to degrees first.
sin 35° is about 0.5736.
cos 55° is about 0.5736 too.
They are the same.
So x = 40 is right!
Let's look back.
First: draw the triangle.
Then: find the other angle.
Then: name the sides from each angle.
Then: write the equation.
Then: remove the + 15.
Last: check.