SOLUTION: 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; Thanks so much in advance:)

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Question 825900: If tanθ = 5/12 and sinθ < 0 , evaluate
a) sinθ
b) cosθ
c) cotθ
d) secθ
e) cscθ
Thanks so much in advance:)
Answer by math-vortex(648)   (Show Source): You can put this solution on YOUR website!
Hi, there--

THE PROBLEM:
If tanθ = 5/12 and sinθ < 0 , evaluate 
a) sinθ
b) cosθ
c) cotθ
d) secθ
e) cscθ 

A SOLUTION:
Let θ 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 sinθ

We are given tanθ = 5/12.

By definition tanθ 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 θ 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 sinθ<0. Since tanθ is positive and sinθ 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

sinθ = y/c (opposite/hypotenuse)
sinθ = -5/13



By similar reasoning:

b) cosθ = x/c = -12/13
c) cotθ = 1/tanθ = 12/5
d) secθ = 1/cosθ = -13/12
e) cscθ = 1/sinθ = -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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