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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
