find tanθ given cos(-θ)=12/13 and θ is in quadrant IV

[Raged]

Find tan(θ), given that cos(-θ) = 12/13 and θ is in quadrant IV

I think I'd know how to do this if the cos(-θ) wasn't negative. So far I have tan(θ)= -5/12 but I haven't changed anything in regards to the negative theta on the cosine. How would the negate change the answer and when? Do I just multiply the -5/12 by a negative?

 

[tkhunny]

I think I'd know how to do this if the cos(-θ) wasn't negative. So far I have tan(θ)= -5/12 but I haven't changed anything in regards to the negative theta on the cosine. How would the negate change the answer and when? Do I just multiply the -5/12 by a negative?

Might want to check out another identity, or two: \(\displaystyle \cos(-\theta) = cos(\theta)\) - That's a good, useful one.

 

[stapel]

Find tan(θ), given that cos(-θ) = 12/13 and θ is in quadrant IV

I think I'd know how to do this if the cos(-θ) wasn't negative.

Look at the cosine wave. What first-quadrant value corresponds to a given fourth-quadrant value? How does this allow you to "simplify" the argument of the cosine?

So far I have tan(θ)= -5/12...

How? For instance, how did you decide that a fourth-quadrant tangent is negative?

...but I haven't changed anything in regards to the negative theta on the cosine.

Then how did you get a "plus theta" (rather than "minus theta") inside your tangent equation?

How would the negate change the answer and when? Do I just multiply the -5/12 by a negative?

What would be the logical basis for this? (In other words, how will you know to do this, or not, when you don't have the answer in the back of the book, to check your work?)

Try using the regular solution method: First, draw the angle in the given quadrant:

place-holder triangle:

    ^ y
    |
    |
    |
----*------------------*---->
    |//////////////////|   x
    |    //////////////|   
    |        //////////|       
    |            //////|         
    |                //|
    |                  *

Clearly, the "minus" comes from the side parallel to the y-axis, and below the x-axis. Cosine is "adjacent over hypotenuse, so label those sides:

place-holder triangle:

    ^ y
    |
    |
    |        12
----*------------------*---->
    |//////////////////|   x
    |    //////////////|   
    |        //////////|       
    |            //////|         
    |        13      //|
    |                  *

Then use the Pythagorean Theorem to find the length of the "opposite" side, noting that, because you're in the fourth quadrant, that vertical side (paralleling the y-axis), must have a negative value. Then read the value of the tangent directly from the sketch.

If you get stuck, please reply showing all of your steps so far. Thank you!