Question 660148
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Hi, there--

The Problem:
The terminal side of angle &#952; in standard position contains the points (-5,-12) give the 
cos &#952;.

Solution:
We can find the cosine of an angle in a right triangle if we know the length of the adjacent 
side and the length of the hypotenuse.

Angle &#952; is in Quadrant III because both x and y are negative. 

Since (-5,-12) is on the terminal side of the angle &#952;, we can draw a right triangle with 
angle &#952; at the origin. 

The hypotenuse of the triangle is the line segment from (0,0) to (-5,-12).

The side adjacent to angle &#952; is the horizontal line segment along the x-axis between 
(0,0) and (-5,0). The directed side length is a=-5 units.

The side opposite angle &#952; is the vertical line segment between (-5,0) and (-5,-12). The 
directed side length is b=-12 units.

Use the Pythagorean Equation to find the length of the hypotenuse c.

a^2 + b^2 = c^2
(-5)^2 + (-12)^2 =c^2
25 + 144 = c^2
169 = c^2
c = 13

cos(&#952;) = [directed length of adjacent side] / [directed length of hypotenuse]
cos(&#952;) = -12/13


If this is unclear, or you still have questions, please email me.

Mrs. Figge
math.in.the.vortex@gmail.com
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