Step by step
The question
Question: find k.
cos k° = sin(k − 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 k° 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 k + y = 90.
3 Name the sides from each angle
Look at side b.
From the k° 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 k°: 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 k° = 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 k° = sin(k − 20)°.
We found cos k° = sin y°.
Between 0° and 90°, each angle has its own sin.
So y = k − 20.
In k + y = 90, put k − 20 in place of y: k + (k − 20) = 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 k terms go together.
k + k = 2k.
Now we have 2k − 20 = 90.
We want k by itself.
Two things are stuck to k: times 2 and − 20.
First k was times 2.
Then − 20.
To take them off, we go backwards.
First we remove the − 20.
Then the times 2.
6 Remove the − 20
To remove − 20, we add 20.
We do it on both sides, so they stay equal.
− 20 and + 20 make zero.
Cross them out.
90 + 20 = 110.
Now we have 2k = 110.
7 Remove the 2
2k means 2 times k.
We want just k.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just k is left.
110 divided by 2 is 55.
So k = 55!
8 Check
Let's check: put k = 55 in the angles.
k − 20 is 35.
Now with the calculator.
Set the calculator to degrees first.
cos 55° is about 0.5736.
sin 35° is about 0.5736 too.
They are the same.
So k = 55 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 − 20.
Then: remove the 2.
Last: check.