SOLUTION: If sin(theta) = k and theta is obtuse, find an expression for tan (90 + theta)

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Question 1141239: If sin(theta) = k and theta is obtuse, find an expression for tan (90 + theta)
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Obtuse means between 90° and 180°, exclusive of 90° and 180°.
So theta is in QII. so we draw an angle with terminal side in QII 
and assume the angle indicated by the red arc is theta:

 

Then we drop a perpendicular (in green) down to the x-axis from the end of the
terminal side of theta.  That makes a right triangle with the x-axis.



We know that the sine of theta is k, which is k/1.  We also know that
the sine is the opposite over the hypotenuse, sow we make the opposite
side (the green side) equal to the numerator of k/1, which is k, and we make the hypotenuse (the terminal side of theta) be 1.  Then we calculate the
adjacent side on the x-axis by the Pythagorean theorem:

c%5E2=a%5E2%2Bb%5E2
1%5E2=a%5E2%2Bk%5E2
1=a%5E2%2Bk%5E2
1-k%5E2=a%5E2
%22%22%2B-sqrt%281-k%5E2%29=a
And since the adjacent side goe to the left of the origin we take
the negative square root,
-sqrt%281-k%5E2%29=a




Now let's draw the angle 90°+theta by adding 90° to theta, indicated
by the blue arrow, and we draw a perpendicular (in red) to the x-axis:



The lower triangle is congruent to the upper one.  Sides that go left
or down are taken negative, so for 90°+theta we have:



Finally, since the tangent is the opposite over the adjacent, we have



Edwin