Trig right triangle solving question How do you know if you should pick tan40 degrees

[land_of_ice]

Trig right triangle solving question How do you know if you should pick tan, cot, etc for the last part?
if b = 2 , A =40 , find a,c, and B
I found all of them except for c , how do you get c
There's a part almost at the end that goes like tan40 degrees = a/2 and Cos 40 degrees = 2/c
Why do you know to pick Cos 40 , why not tan40 or cot of 40 for that last part there?

 

[Stephen Tashi]

What "last part there"? You haven't included whatever you're talking about in your post.

A guess about the use of cosine: To find an unknown with a trig function, you must have an equation that has the unknown in it along with other known information. If you want to find c and cosine involves c, use cosine.

There is often more than one way to solve for an unknown. If c is the hypotenuse you can also use $$c^2 = a^2 + b^2$$ to solve for it after you find b.

 

[sjb-2812]

Trig right triangle solving question How do you know if you should pick tan, cot, etc for the last part?
if b = 2 , A =40 , find a,c, and B
I found all of them except for c , how do you get c
There's a part almost at the end that goes like tan40 degrees = a/2 and Cos 40 degrees = 2/c
Why do you know to pick Cos 40 , why not tan40 or cot of 40 for that last part there?

Can you upload a picture that explains this?

My guess is, although Stephen Tashi is right, it's usually best to only use values given in a question if you can, just in case your values calculated in other parts of the question are wrong, which could cause confusion if you get odd answers later on.

 

[HallsofIvy]

The convention is that, in a triangle, sides labeled a, b, c are opposite angles labeled A, B, C respectively.

And, in right triangles, by convention, C is the right angle and c is the hypotenuse.

Back when you first learned about trig functions you were supposed to have learned something like:
sine= opposite side/hypotenuse,
cosine= near side/hypotenuse
tangent = opposite side/near side

Given angle A and side b, you think- a is the side opposite angle A and b is the other leg, the side "near" angle A so the appropriate formula is "tangent= opposite side/near side": tan(A)= a/b or tan(40)= a/2. Then solve for a.

To find c, given only angle a and angle b, you think, c is the hypotenuse and b is the "near" side to A so the appropriate formula is "cosine= near side over hypotenuse":
cos(A)= b/c or cos(40)= 2/c. Then solve for c.

Of course, if, at this point, you have already solved for a, you could think "sine= opposite side over hypotenuse": sin(40)= a/c. Or, as Stephen Tashi said, you could use \(\displaystyle c^2= a^2+ b^2\). However, I agree with sjb-2812 that it is better to use the initially given values- you will not propagate arthmetic errors.