Trigonometric functions and the unit circle

In summary, the conversation was about finding the solutions to the equation \cos \theta = -1/2 and writing them as a function of an integer n. One suggestion was to write the solutions as \theta = \frac{2\pi}{3} + 2 \pi n, \frac{4 \pi}{3} + 2 \pi n. Another suggestion was to write them as \frac{2 \epsilon \pi}{3} + 2 \pi n with n \in \mathbb{Z}, \epsilon = 1, 2 or \frac{k \pi}{3} with k \in \mathbb{Z}, k \equiv 2 \text{ or } 4
  • #1
Niles
1,866
0

Homework Statement


Hi all.

Today I had to solve: [itex]\cos \theta = -1/2[/itex]. What I did was to look in a table to find that [itex]\theta = 2\pi/3 \quad \text{and}\quad \theta = 4\pi/3[/itex].

My question is what is the general strategy when I wish to write this as a a function of an integer n? Is there even a general strategy for this?

Thanks in advance.
 
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  • #2
Niles said:
... when I wish to write this as a a function of an integer n? Is there even a general strategy for this?

What do you mean?
Can you solve
[tex]\cos\theta = n[/tex]
without looking it up (no, unless you have a calculator), or can you write the solutions
[tex]\theta = 2\pi/3, 4 \pi / 3 [/tex]
as
[tex]\theta = \frac{2\pi}{3} + 2 \pi n, \frac{4 \pi}{3} + 2 \pi n[/tex]
or can you combine those into one formula (no, not particularly nicely).
 
  • #3
What I meant was your #2 suggestion: If there is any way to write the two solutions nicely for all n.

Hmm, ok then. I will just stick to writing the solutions as you did. Thanks.
 
  • #4
OK, so maybe you can write something like
[tex]\frac{2 \epsilon \pi}{3} + 2 \pi n[/tex]
with [itex]n \in \mathbb{Z}, \epsilon = 1, 2[/itex]
or
[tex]\frac{k \pi}{3}[/tex]
with [itex]k \in \mathbb{Z}, k \equiv 2 \text{ or } 4 \, \operatorname{mod} 7[/itex].

But I think the expression from my earlier post is more common (in any case, you only need to think about it half as long to understand what it's saying and where it comes from :smile:).
 
  • #5
Ok, thanks!
 

Related to Trigonometric functions and the unit circle

1. What are the six basic trigonometric functions?

The six basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.

2. How do you find the values of trigonometric functions on the unit circle?

To find the values of trigonometric functions on the unit circle, you can use the coordinates of the point where the angle intersects the unit circle. For example, the sine value is the y-coordinate and the cosine value is the x-coordinate of the point.

3. What is the relationship between trigonometric functions and right triangles?

Trigonometric functions are defined as ratios of the sides of a right triangle. The sine of an angle is equal to the ratio of the opposite side to the hypotenuse, the cosine is equal to the adjacent side to the hypotenuse, and the tangent is equal to the opposite side to the adjacent side.

4. How do you use the unit circle to solve trigonometric equations?

You can use the unit circle to solve trigonometric equations by substituting the values of the angles in the equation with the corresponding coordinates on the unit circle. This can help you find the values of the trigonometric functions for that angle.

5. What is the purpose of the unit circle in trigonometry?

The unit circle is used in trigonometry as a way to visualize and understand the relationship between trigonometric functions and angles. It also simplifies calculations and can be used to solve trigonometric equations.

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