Question 825900
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Hi, there--

THE PROBLEM:
If tan&#952; = 5/12 and sin&#952; < 0 , evaluate 
a) sin&#952;
b) cos&#952;
c) cot&#952;
d) sec&#952;
e) csc&#952; 

A SOLUTION:
Let &#952; be the measure the acute angle A in the right triangle ABC.
Let c be the length of the hypotenuse.
Let x be the length of the leg adjacent to angle A.
Let y be the length of the leg opposite angle A. 

a) Find sin&#952;

We are given tan&#952; = 5/12.

By definition tan&#952; is the ratio of the side opposite to the side adjacent, or y/x=5/12. 

We can use the Pythagorean Equation to find the length of the hypotenuse c when side lengths 
are exactly 5 and 12. The ratios will be the same for any right triangle with acute 
angle &#952; because they will be similar triangles. 

5^2 + 12^2 = c^2
c^2 = 25 + 144
c^2 = 169
c = 13



Recall that we have the constraint that sin&#952;<0. Since tan&#952; is positive and sin&#952; is negative, this 
corresponds to an angle in Quadrant III (x<0. y<0) of the unit circle. We need to adjust signs for the trig functions accordingly.

x = -12
y=-5
c=13

sin&#952; = y/c (opposite/hypotenuse)
sin&#952; = -5/13



By similar reasoning:

b) cos&#952; = x/c = -12/13
c) cot&#952; = 1/tan&#952; = 12/5
d) sec&#952; = 1/cos&#952; = -13/12
e) csc&#952; = 1/sin&#952; = -12/5
 

Hope this helps! Feel free to email if you have any questions.

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