Step by step
The question
Question: find x.
cos(3x)° = sin(x + 14)°.
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 (3x)° 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 3x + y = 90.
3 Name the sides from each angle
Look at side b.
From the (3x)° 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 (3x)°: 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 (3x)° = 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(3x)° = sin(x + 14)°.
We found cos(3x)° = sin y°.
Between 0° and 90°, each angle has its own sin.
So y = x + 14.
In 3x + y = 90, put x + 14 in place of y: 3x + (x + 14) = 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.
3x + x = 4x.
Now we have 4x + 14 = 90.
We want x by itself.
Two things are stuck to x: times 4 and + 14.
First x was times 4.
Then + 14.
To take them off, we go backwards.
First we remove the + 14.
Then the times 4.
6 Remove the + 14
To remove + 14, we take away 14.
We do it on both sides, so they stay equal.
+ 14 and − 14 make zero.
Cross them out.
90 − 14 = 76.
Now we have 4x = 76.
7 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
76 divided by 4 is 19.
So x = 19!
8 Check
Let's check: put x = 19 in the angles.
3x is 57.
x + 14 is 33.
Now with the calculator.
Set the calculator to degrees first.
cos 57° is about 0.5446.
sin 33° is about 0.5446 too.
They are the same.
So x = 19 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 + 14.
Then: remove the 4.
Last: check.