Step by step
The question
Question: find k.
cos 25° = sin k°.
k is 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 25° at the bottom.
The other small angle is at the top.
We do not know it yet.
2 Find the other angle
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°.
One small angle is 25°.
The other one gets the rest: 90 minus 25.
90 minus 25 is 65.
So the other angle is 65°.
3 Name the sides from each angle
Look at side b.
From the 25° 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 25°: 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 25° = sin 65°.
This works in every right triangle.
The cos of one small angle is the sin of the other.
Together they make 90°.
4 Find k
The question says cos 25° = sin k°.
We found cos 25° = sin 65°.
So sin k° = sin 65°.
Careful!
cos and sin of the same angle are not equal.
cos 25° is about 0.9063, but sin 25° is about 0.4226.
Between 0° and 90°, each angle has its own sin.
So k = 65.
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
cos 25° is about 0.9063.
sin 65° is about 0.9063 too.
They are the same.
So k = 65 is right!
Let's look back.
First: draw the triangle.
Then: find the other angle.
Then: name the sides from each angle.
Then: find k.
Last: check.