Step by step
The question
Question: find y.
sin(4y − 10)° = cos(2y + 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 (4y − 10)° at the bottom.
The other small angle is at the top.
We call it z.
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 4y − 10 + z = 90.
3 Name the sides from each angle
Look at side a.
From the (4y − 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 (4y − 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 (4y − 10)° = cos z°.
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(4y − 10)° = cos(2y + 10)°.
We found sin(4y − 10)° = cos z°.
Between 0° and 90°, each angle has its own cos.
So z = 2y + 10.
In 4y − 10 + z = 90, put 2y + 10 in place of z: 4y − 10 + (2y + 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 y terms go together.
Plain numbers go together.
4y + 2y = 6y.
−10 + 10 = 0.
Now we have 6y = 90.
6 Remove the 6
6y means 6 times y.
We want just y.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just y is left.
90 divided by 6 is 15.
So y = 15!
7 Check
Let's check: put y = 15 in the angles.
4y − 10 is 50.
2y + 10 is 40.
Now with the calculator.
Set the calculator to degrees first.
sin 50° is about 0.766.
cos 40° is about 0.766 too.
They are the same.
So y = 15 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 6.
Last: check.