Plz explain reference angle and help solve this problem

[Kamhogo]

Question: tan theta = -0.2126. Find theta to 4 significant figures for 0<\=theta= < 2 pi


I found tan^(-1)(-0.2126) = -0.2095 rads, then I found theta = 2 pi - 0.2095= 6.074 rada.

I don't really understand what I'm doing, just trying multiple ways. Can some please explain
what is asked of me and how to deliver?

The textbook answers are 6,074 rads(which I had correct by chance) and 2.932 rads ( which I
can't get too). Any help is welcome.

 

[ksdhart]

The thing to remember is the tan^-1 button on your calculator returns only the principle value of the inverse tangent. Because tangent is a periodic function, you can add (or subtract) any multiple of the period to theta and the tan(theta) will have the same value. So, do you recall what the period of tangent is? I'm sure you can then see how the two answers from the textbook came about.

 

[Kamhogo]

Is the period of tan pi?
I added the the inverse tangent of -0.2126 to pi and 2pi and got the correct answers but I still don't understand the reasoning behind it all. Can you please explain some more?
Tan (theta ) = - 0.2126
Inverse tan (-0.2126) = -0.2095
2pi - 0.2095 = 6.074
Pi - 0.2025 = 2.932

 

[Kamhogo]

I have to solve a similar problem and I can't for the life of me: "Find theta to 4 significant figures for 0 <\= theta < 2pi.

Sec (theta ) = - 1.307"

Attempt at solution :
Inverse sec (-1.307) = 2,442

Period of = 2 pi
Secant is negative in 2nd and 3rd quadrants.
Sec (2pi- 2.442) = -1.307
Stuck from here. Please help

 

[Kamhogo]

Answers female the textbook: 0.8309 and 5.452

 

[pka]

Answers female the textbook: 0.8309 and 5.452

What in the world does that mean? Are you having trouble with translations?

Question: tan theta = -0.2126. Find theta to 4 significant figures for 0<\=theta= < 2 pi
I found tan^(-1)(-0.2126) = -0.2095 rads, then I found theta = 2 pi - 0.2095= 6.074 rada.
I don't really understand what I'm doing, just trying multiple ways. Can some please explain.

Having read this entire thread, it is perfectly that either none of it clicks with you or you are making too much of it.

You need to follow a basic pattern. Suppose that \(\displaystyle \tan(\theta)=X\) then
If \(\displaystyle X>0\) then \(\displaystyle \theta=\arctan(X)\text{ or else }\theta=\arctan(X)+\pi\) that gives values in I or III.
If \(\displaystyle X<0\) then \(\displaystyle \theta=\arctan(X)+\pi\text{ or else }\theta=\arctan(X)+2\pi\) that gives values in II or IV.

If you simply work with those two rules, one for positive, one for negative, you will become comfortable with it.

 

[ksdhart]

My apologies if I was unclear before. I'll see if I can explain a bit better. For the first problem you posted, let's look at what we know. You're asked to find all values of theta, between 0 and 2pi, that satisfy this equation:

\(\displaystyle tan(\theta)=-0.2126\)

One possible value of theta is given by:

\(\displaystyle \theta=tan^{-1}(-0.2126)=-0.2095\)

Rewriting the above gives us:

\(\displaystyle tan(-0.2095) = -0.2126\)

Tangent, like all the other trigonometric functions is periodic. It repeats infinitely with period pi. So, let's add some multiples of pi to theta and see what happens:

\(\displaystyle tan(-0.2095+\pi)=tan(2.93209)=-0.2126\)

\(\displaystyle tan(-0.2095+2\pi)=tan(6.07369)=-0.2126\)

Thus, we can see that there are two angles, within the given bounds, whose tangent has the given value. Thus, those angles are our two solutions for theta.

For your second problem, I don't agree with your textbook's answers at all. You can plainly check them for yourself and see that they're wrong. Your logic and solutions, however, seem fine to me.

\(\displaystyle sec(0.8309)=1.483\ne-1.307\)

\(\displaystyle sec(5.4523)=1.483\ne-1.307\)