Step by step
The question
Question: find x.
sin(x + 10)° = cos(2x + 20)°.
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 + 10)° 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 + 10 + y = 90.
3 Name the sides from each angle
Look at side a.
From the (x + 10)° 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 (x + 10)°: 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 (x + 10)° = cos y°.
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(x + 10)° = cos(2x + 20)°.
We found sin(x + 10)° = cos y°.
Between 0° and 90°, each angle has its own cos.
So y = 2x + 20.
In x + 10 + y = 90, put 2x + 20 in place of y: x + 10 + (2x + 20) = 90.
Careful!
The two angles are not equal.
sin and cos 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.
10 + 20 = 30.
Now we have 3x + 30 = 90.
We want x by itself.
Two things are stuck to x: times 3 and + 30.
First x was times 3.
Then + 30.
To take them off, we go backwards.
First we remove the + 30.
Then the times 3.
6 Remove the + 30
To remove + 30, we take away 30.
We do it on both sides, so they stay equal.
+ 30 and − 30 make zero.
Cross them out.
90 − 30 = 60.
Now we have 3x = 60.
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.
60 divided by 3 is 20.
So x = 20!
8 Check
Let's check: put x = 20 in the angles.
x + 10 is 30.
2x + 20 is 60.
Now with the calculator.
Set the calculator to degrees first.
sin 30° is 0.5.
cos 60° is 0.5 too.
They are the same.
So x = 20 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 + 30.
Then: remove the 3.
Last: check.