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Sine and Cosine Stating Negative Angles
Let’s take a look at this pic below
Now as the angle x goes down and down toward 0, the side on the front of the angle also get shorten and shorten. When x goes negative the side in front of the angle point downward.
The side near the angle remain constant though.
As x goes down to exactly -x we got a congruent triangle except flipped outside down.
So the cosine remain the same, namely b/c. The sine is just the additive inverse. Before it’s a/c, now it’s -a/c.
Hence, we easily see sin(-x)=-sin(x) and cos(-x)= cos(x)
Sine and Cosine Stating Negative Numbers the Radians 2
Now look at this wikipedia pics again that I plagiarized.
I am too lazy to draw my own work and lazy is good.
our cosine function has a little change of definition. Rather than the ratio between the length of the near side to the tilted side, now it’s how far to the right the near side is.
Now from the pics, it’s obvious that
cos(θ)=cos(-θ)
or I can more conveniently type
cos(x)=cos(-x)
Reversing that angle do not change the cosine.
Sine and Cosine Stating Negative Numbers the Radians 1
Now I just took some pics from wikipedia. That’s what happen when people are nice. Everybody becomes freeloaders.
Now smart ass mathematician often state angles in term of radians rather than degree. Radians are ratio between the length of the curve in front of the angel to the radius of the circle.
Because there are 360° of angle in the circle, and the circumference of a circle is 2π it’s radius, we would expect 360° to be equivalent with 2π radians. Radians are dimensionless being just ratio.
So 180° is just 180°/360°*2π=π
Simple?