Why Does cos(180-Θ) = -cos(Θ)?

[teetar]

1. Explain, using a unit circle diagram, why cos(180-Θ) = -cos(Θ)

2. My Attempt at a Solution:
I have no idea how to begin thinking of this. I don't know any of the obvious details of the question, the only rational thought I can currently form is that cos(Θ) = the x-coordinate of the terminal angle's intersection with a unit circle, however, I don't even know if that's related to the question. I cannot form any simple thoughts regarding this question. I have no idea where to begin, so I'm just wondering if someone here might be able to say something that could point me in the correct direction of an answer. Thanks!

 

[jahaan]

Draw a circle... Think about how you can indicate cosine on the circle. You already know the definition of cosine, so you are on the right track...

 

[LCKurtz]

Or if you have had the addition formulas you can use the formula for $\cos(a-b)$.

 

[adjacent]

Or if you have had the addition formulas you can use the formula for $\cos(a-b)$.

The question asked to explain using a unit circle diagram.

 

[verty]

Look here...

 

[acegikmoqsuwy]

If you look at the unit circle, you can notice that the x-values are "reflective" of each other in the top half. This is because at the $90\deg$ point, $x=0$ so when you decrease the angle, $x$ increases the same amount as it would decrease if you had rather increased the angle. From this observation, you can make the statements that $\cos(90-\theta)=x$ and $\cos(90+\theta)=-x$ Now, do what you will with those statements and the relation $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$

 

[HakimPhilo]

Here's an image I made (sorry for the bad quality) that will show you the intuition behind that formula:

$$\color{#50F}{\cos(\pi-\theta)}=-\color{fuchsia}{\cos(\theta)}.$$

 

[NascentOxygen]

"unit circle" means its radius is 1. So sine = opp/hyp = opp/1 = opp

 

[FilupSmith]

What you have to remember is that when you draw your unit circle, you need to remember ASTC.

The region which 180-θ falls in is the second quadrant — or, the S quadrant. In this quadrant, only sin is positive. So cos(180-θ)=-cosθ

Hope this helps ~| FilupSmith |~