Step by step
The question
Question: find x.
sin x° = cos(3x − 10)°.
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° 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 + y = 90.
3 Name the sides from each angle
Look at side a.
From the x° 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°: 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° = 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° = cos(3x − 10)°.
We found sin x° = cos y°.
Between 0° and 90°, each angle has its own cos.
So y = 3x − 10.
In x + y = 90, put 3x − 10 in place of y: x + (3x − 10) = 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.
x + 3x = 4x.
Now we have 4x − 10 = 90.
We want x by itself.
First we remove the − 10, so 4x is alone.
Then we remove the times 4.
Is x alone now?
Not yet: − 10 is still with x.
How do we remove it?
6 Remove the − 10
To remove − 10, we add 10.
We do it on both sides, so they stay equal.
− 10 and + 10 cancel each other.
They make zero.
90 + 10 = 100.
Now we have 4x = 100.
So after step 6: we added 10, and now we have 4x = 100.
Is x alone now?
Not yet: 4x means 4 times x.
7 Remove the 4
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
They cancel each other.
So just x is left.
100 divided by 4 is 25.
So x = 25!
8 Check
Let's check: put x = 25 in the angles.
3x − 10 is 65.
Now with the calculator.
Set the calculator to degrees first.
sin 25° is about 0.4226.
cos 65° is about 0.4226 too.
They are the same.
So x = 25 is right!
9 Type it in
The answer box takes 5 characters.
25 has 2.
It fits.
We type 25.
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 − 10.
Then: remove the 4.
Then: check.
Last: type it in.