Step by step
The question
Question: find x.
cos(x + 30)° = sin(2x + 60)°.
Both angles are 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 (x + 30)° at the bottom.
The other small angle is at the top.
We call it y.
2 The two small angles
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°.
So x + 30 + y = 90.
3 Name the sides from each angle
Look at side b.
From the (x + 30)° 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 (x + 30)°: 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 (x + 30)° = sin y°.
This works in every right triangle.
The cos of one small angle is the sin of the other.
Together they make 90°.
4 Write the equation
The question says cos(x + 30)° = sin(2x + 60)°.
We found cos(x + 30)° = sin y°.
Between 0° and 90°, each angle has its own sin.
So y = 2x + 60.
In x + 30 + y = 90, put 2x + 60 in place of y: x + 30 + (2x + 60) = 90.
Careful!
The two angles are not equal.
cos and sin are different.
The angles add up to 90.
A plus in front of a bracket: we can just drop the bracket.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
x + 2x = 3x.
30 + 60 = 90.
Now we have 3x + 90 = 90.
We want x by itself.
Two things are stuck to x: times 3 and + 90.
First x was times 3.
Then + 90.
To take them off, we go backwards.
First we remove the + 90.
Then the times 3.
6 Remove the + 90
To remove + 90, we take away 90.
We do it on both sides, so they stay equal.
+ 90 and − 90 make zero.
Cross them out.
90 − 90 = 0.
Now we have 3x = 0.
7 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
0 divided by 3 is 0.
So x = 0!
8 Check
Let's check: put x = 0 in the angles.
x + 30 is 30.
2x + 60 is 60.
Now with the calculator.
Set the calculator to degrees first.
cos 30° is about 0.866.
sin 60° is about 0.866 too.
They are the same.
So x = 0 is right!
Let's look back.
First: draw the triangle.
Then: the two small angles.
Then: name the sides from each angle.
Then: write the equation.
Then: group like terms.
Then: remove the + 90.
Then: remove the 3.
Last: check.