How to solve for theta for this trig question?

In summary, to solve this problem, one must use the Pythagorean theorem to find the opposite length, use the relationships between different trigonometric functions, and use a calculator or inverse trigonometric functions to determine the exact value of each trigonometric ratio. It is also important to consider the quadrant in which theta lies to get the correct answer.
  • #1
zeion
466
1

Homework Statement



cos(theta) = -3/4, pi <= theta <= 2pi

Homework Equations





The Attempt at a Solution



I forgot how to do this. How do I use the special triangles to do this question? Do I need to square the sqrt(3) / 2 one or something? Thanks.
 
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  • #2
Try the Pythagorean theorem.
 
  • #3
So I use Pythagorean to get the opposite length? (sqrt(7) / 4) ?
 
  • #4
zeion said:
So I use Pythagorean to get the opposite length? (sqrt(7) / 4) ?
It depends what you're trying to find. That will help if you want some other trig function of theta, but not much use if you want theta itself.
If it's a numerical value for theta that you want, at some point you will have to use a calculator or whatever to compute an arccos. But the more interesting aspect of the question is getting an answer in the right quadrant. Do you know the relationships between cos(x), cos(-x), cos(pi+x), etc?
 
  • #5
Not all trig functions require the use of a calculator to evaluate.
 
  • #6
SteamKing said:
Not all trig functions require the use of a calculator to evaluate.
True, but certainly to find arccos in this case.
 
  • #7
Not really. If you know basic trig values for 45-45-90 and 30-60-90 triangles, solving this problem should be possible without a calculator.
 
  • #8
SteamKing said:
Not really. If you know basic trig values for 45-45-90 and 30-60-90 triangles, solving this problem should be possible without a calculator.
For |cos(theta)| = 3/4? Are you sure about that? If 3/5 I'd agree.
 
  • #9
I'm pretty sure that zeion has left the building ...

At any rate, given a value for cos(θ), for π ≤ θ ≤ 2π, one can determine in which quadrant θ lies, and thus determine the values of all the other trig functions at angle θ.

Of course, we're still not sure about just what zeion was supposed to do in regards to this problem.
 
  • #10
SammyS said:
I'm pretty sure that zeion has left the building ...

At any rate, given a value for cos(θ), for π ≤ θ ≤ 2π, one can determine in which quadrant θ lies, and thus determine the values of all the other trig functions at angle θ.

Of course, we're still not sure about just what zeion was supposed to do in regards to this problem.

I'm supposed to

i) Sketch the angle and state the related acute angle
ii) Determine the exact value of each trigonometric ratio

The previous 3 questions I could do because they were all values I could find on the special triangle so they were easy. But I'm not sure how to relate 3/4 to the special triangles?
 
  • #11
Since [itex]\theta[/itex] lies between [itex]\pi[/itex] and [itex]2\pi[/itex], and [itex]cos(\theta)[/itex] is negative, [itex]\theta[/itex] is in the fourth quadrant. Use a calculator to find [itex]cos^{-1}[/itex] of -4/3 and add [itex]\pi[/itex]. Equivalently, find [itex]cos^{1}[/itex] of 3/4 and subtract from [itex]2\pi[/itex].
 
  • #12
HallsofIvy said:
Since [itex]\theta[/itex] lies between [itex]\pi[/itex] and [itex]2\pi[/itex], and [itex]cos(\theta)[/itex] is negative, [itex]\theta[/itex] is in the fourth quadrant. Use a calculator to find [itex]cos^{-1}[/itex] of -4/3 and add [itex]\pi[/itex]. Equivalently, find [itex]cos^{1}[/itex] of 3/4 and subtract from [itex]2\pi[/itex].
That should be, θ is in the third quadrant.

cos(θ) is positive for θ in the first & fourth quadrants.

cos(θ) is negative for θ in the second & third quadrants.
 
Last edited:
  • #13
It is possible that a numerical answer is not required. An expression involving 'arccos(3/4)' may be acceptable. The trick is to get the expression right. arccos is defined to return a value in [0, pi), but the required answer is in [pi, 2pi].
 

1. How do I find the value of theta in a trigonometry problem?

To solve for theta in a trigonometry problem, you need to use the given information and the trigonometric ratios (sine, cosine, and tangent) to set up an equation. Then, you can use algebraic techniques to isolate theta and solve for its value.

2. What are some common strategies for solving for theta in a trigonometry problem?

One common strategy is to draw a diagram and label all the given information, including the sides and angles of the triangle. Another strategy is to use the Pythagorean theorem to find missing side lengths, if necessary. You can also use the unit circle or memorize common trigonometric values to simplify calculations.

3. What should I do if the problem involves multiple angles and trigonometric functions?

If the problem has multiple angles and functions, you will need to use trigonometric identities to simplify the expressions and solve for theta. These identities include the sum and difference identities, double angle identities, and half angle identities.

4. Can I use a calculator to solve for theta in a trigonometry problem?

Yes, you can use a calculator to solve for theta in a trigonometry problem. However, it is important to know how to solve for theta using algebraic techniques as well, as some problems may not be able to be solved using a calculator.

5. What should I do if I encounter a trigonometry problem with no given information?

If there is no given information in the problem, you will need to use trigonometric identities or other known values (such as the length of the hypotenuse in a right triangle) to set up an equation and solve for theta. You may also need to make assumptions or use logical reasoning to determine the missing information.

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