Step by step
The question
Question: find sin Q.
Triangle PQR has a right angle at R, and cos P = 5/13.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw the triangle
The right angle is at R.
So P and Q are the two small angles of one right triangle.
We draw it: P at the bottom, Q at the top.
2 Name the sides from each angle
Look at side b.
From P, it is next to it: the adjacent side.
We write opp, adj and hyp for short.
From Q, the same side b is across: the opposite side.
Side c is the hypotenuse for both angles.
cos is the adjacent side over the hypotenuse.
From P: b over c.
sin is the opposite side over the hypotenuse.
From Q: b/c too.
Careful!
Q is the other angle.
The side across from Q is the side next to P.
The right way: sin Q = opp over hyp = b over c.
Same two sides, same fraction.
So sin Q = cos P.
3 Find sin Q
cos P is 5/13.
So sin Q = 5/13 too.
4 Check
Let's check with the calculator.
The cos⁻¹ button turns cos P back into the angle: P is about 67.4°.
So Q is 90 minus 67.4: 22.6°.
sin 22.6° is about 0.38.
That is 5/13 again: 5 divided by 13 is 0.3846.
So sin Q = 5/13 is right!
5 Pick the choice
Our answer is sin Q = 5/13.
That is choice B.
6 Why not the others
A is 5/12.
Wrong: that is tan P: opposite over adjacent.
C is 12/13.
Wrong: that is sin P: Q is the other angle.
D is 13/5.
Wrong: upside down: sin is never more than 1.
Let's look back.
First: draw the triangle.
Then: name the sides from each angle.
Then: find sin Q.
Then: check.
Then: pick the choice.
Last: why not the others.