exact value of cos (-75 degrees)

[ru3ful]

Hi there, I understand cos(75) fine, but this just does not make ANY sense to me? Help please!

 

[MarkFL]

Think of two points on the unit circle both initially at (1,0), i.e., \(\displaystyle \theta=0\). One particle moves in the positive angular direction, and the other in the negative direction, both at the same angular speed. How is the \(\displaystyle x\)-coordinate of the two particles related, that is, how are \(\displaystyle \cos(\theta)\) and \(\displaystyle \cos(-\theta)\) related?

 

[HallsofIvy]

Hi there, I understand cos(75) fine, but this just does not make ANY sense to me? Help please!

Okay, what do you understand "cos(75)" to be? There are a number of different ways of "defining" or "visualizing" cosine. Which are you using?

MarkFL mentions the "unit circle" definition, in which we measure a distance, t, around the circumference of a unit circle, counterclockwise for positive t, clockwise for negative t, (If t is in radians. Use the central angle if in degrees), starting at (1, 0) and ending at a given point (x, y) on the circle. "cos(t)" is defined to be the x coordinate of that point. Since x will be decreasing whether we go counterclockwise or clockwise, we have cos(-t)= cos(t) for any t.

Another method, a little more closely related to the elementary "right triangle" definitions, is to draw a right triangle with one leg along the x axis, the other perpedicular to the x-axis and the hypotenuse from (0, 0). If t is positive, the vertical leg is upward, if negative, downward. As long as t is between -90 and 90 degrees, the "near leg", along the x-axis, is positive, so cos(t), "near side over hypotenuse", is postive whether t is positive or negative. Again, cos(-t)= cos(t).

 

[Deleted member 4993]

Another way:

cos(-75°)

= cos(75°)

= cos(30° + 45°)

= cos(30°)*cos(45°) - sin(30°)*sin(45°)

Continue.......

 

[pka]

Hi there, I understand cos(75) fine, but this just does not make ANY sense to me? Help please!

You say
I understand \(\displaystyle \cos(75^o)\)!

So why don't you know the basics: \(\displaystyle \cos(-\theta)=\cos(\theta)~?\)

OR i.e, the cosine is an even function.