Trig. Q: If cosθ = 0.7, find the value of cos(θ - 4π) + cos(θ + 2π)

[NewStudent]

Trig. Q: If cosθ = 0.7, find the value of cos(θ - 4π) + cos(θ + 2π)

"If cosθ = 0.7, find the value of cos(θ - 4π) + cos(θ + 2π). Justify your answer."

We have never talked about how to do this in class, and I am incredibly lost. How do I plug in 0.7 if cos and θ are separated in the problem, yet they need to be together to equal 0.7? Am I missing something?

 

[Otis]

"If cosθ = 0.7, find the value of cos(θ - 4π) + cos(θ + 2π). Justify your answer."

We have never talked about how to do this in class, and I am incredibly lost. How do I plug in 0.7 if cos and θ are separated in the problem, yet they need to be together to equal 0.7? Am I missing something?

Your question, "How do I ..." leads me to think that you're missing the concepts of function notation and period, but I'm not sure.

cos(θ) = 0.7

The expression in blue is function notation; it's a symbol which stands for the output of the cosine function.

In other words, "cos" is the name being used for the function, in this notation, and symbol θ represents the input value.

[Similar to the notation f(x); the name of the function is "f" and x represents the input; the symbol f(x) represents the output.]

To say cos(θ)=0.7 means that, when some specific value θ goes into the cosine function, the value that comes out is 7/10.

We don't need to know what that specific value of θ is, in this exercise. We do need to know that when this particular value of θ (whatever it is) goes into the cosine function, the value 7/10 comes out.


This is also function notation: cos(θ - 4π).

The expression θ - 4π represents the input (it's 4Pi units smaller than θ), and the entire symbol cos(θ - 4π) represents the output.

Same goes for cos(θ + 2π). If we input the value which is 2Pi units larger than θ, then cos(θ + 2π) comes out.


Have you learned the concept in trigonometry known as "period" ?

Trig functions are periodic; their outputs repeat a pattern over and over, as the inputs continuously increase or decrease. For the sine and cosine functions, this pattern looks like a wave. (You've seen graphs of sine and cosine functions, yes?) This periodic nature means that there are an infinite number of input values that ALL result in the same output.

The period of cosine is a number. Graphically speaking, the period is the "width" of one complete wave. Or, said another way, if you're at one point on the wave, and you want to get to the same point on the next wave, the period is the amount by which the input must increase. The period of cosine is what allows us to determine the values of cos(θ - 4π) and cos(θ + 2π), without knowing θ.

Please let us know whether you've learned the period for cosine, yet. If you have, tell us what it is, and we can go from there. If you have not, then try using the index in your textbook to find examples. The concept of period and its specific value for the cosine function are crucial to understanding and completing this exercise.

Cheers

 

[pka]

"If cosθ = 0.7, find the value of cos(θ - 4π) + cos(θ + 2π). Justify your answer."

This whole exercise depends upon: \(\displaystyle \large\cos(\theta\pm\eta)=\cos(\theta)\cos(\eta) \mp\sin(\theta)\sin(\eta)\) where \(\displaystyle \eta=2n\pi\)
\(\displaystyle \cos(2n\pi)=1~\&~\sin(2n\pi)=0\)