Question 849814: what is the value of sin(A+B) if 5cos A-4=0 and 12tan B+5=0?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 5cosA-4 = 0
add 4 to both sides of this equation to get:
5cosA = 4
divide both sides of this equation by 5 to get:
cosA = 4/5
12tanB+5 = 0
subtract 5 from both sides of this equation to get:
12tanB = -5
divide both sides of this equation by 12 to get:
tanB = -5/12
when the cosine of angle A is positive, angle A can be in either the first quadrant or the fourth quadrant.
when the tangent of angle B is negative, angle B can be in either the second quadrant or the fourth quadrant.
i usually find it easiest to find the angle in the first quadrant and then do the transposition from there.
if the angle is in the first quadrant then the cosine and the tangent will both be positive.
angle A in the first quadrant is equal to arccos (4/5) which is equal to 36.86989765 degrees.
angle B in the first quadrant is equal to arctan (5/12) which is equal to 22.61986495.
now that the angle has been found in the first quadrant, it is relatively simple to find the angle in other quadrants.
The cosine of an angle is positive when the angle is in the first quadrant or the fourth quadrant
in the first quadrant, angle A is equal to 36.86989765 degrees.
in the fourth quadrant, angle A is equal to 360 - 36.86989765 degrees which is equal to 323.1301025 degrees.
to verify this is accurate, use your calculator to find the cosine of 36.86989765 degrees and the cosine of 323.1301025 degrees.
you should find that both cosines are positive and are equal to 4/5.
the tangent of an angle is negative when the angle is in the second quadrant and the fourth quadrant.
in the second quadrant, angle B is equal to 180 - 22.61986495 degrees which is equal to 157.3801351 degrees.
in the fourth quadrant, angle B is equal to 360 - 22.61986495 degrees which is equal to 337.3801351 degrees.
to verify this is accurate, use your calculator to find the tangent of 157.3801351 degrees and the tangent of 337.3801351 degrees.
you should find that both tangents are negative and are equal to 5/12.
i checked with my calculator and it confirms that these values are good.
you are not done yet though.
you want to find sin(A+B).
you need to add the angles together and then find the sine.
since A is in the first quadrant and the fourth quadrant, and B is in the second quadrant and the fourth quadrant, you have 2 * 2 = 4 possible combinations.
they are:
A in the first quadrant plus B in the second quadrant.
A in the first quadrant plus B in the fourth quadrant.
A in the fourth quadrant plus B in the second quadrant.
A in the fourth quadrant plus B in the fourth quadrant.
here's where it gets a little murky for me.
i believe you want to find the quadrant that both angles are in.
this allows you to get one solution.
both angles are in the fourth quadrant.
in the fourth quadrant, sine is negative and cosine is positive and tangent is negative.
the fourth quadrant satisfies the requirement that cosine (A) is positive and tangent (B) is negative.
i believe you want to find sin(A+B) in the fourth quadrant.
in the fourth quadrant,the sine of the angle is negative.
angle A is equal to 323.1301024 degrees.
angle B is equal to 337.3801351 degrees.
add these 2 angles together and you get an angle of 660.5102374 degrees.
use your calculator to find the sine of this angle and you get:
sin(660.5102374) = -.8615384615.
if my assumption is correct, that's your answer.
you could also have gotten the equivalent angle less than 360 degrees by subtracting 360 from 660 to get angle (A+B) = 300.5102374 degrees.
sin(300.5102374) = -.8615384615
this is to be expected since we are talking about the same angle.
i also looked at both angles as if they were in the first quadrant and i got an angle of 59.48976259 that produced a sine of .8615384615.
take that angle of 59.48976259 degrees and transpose it into the fourth quadrant and you get an angle of 360 - 59.48976259 degrees which is equal to 300.5102374 degrees.
an angle of 300.5102374 degrees is also equivalent to an angle of -59.48976259 degrees.
I'm pretty sure I'm correct.
I'm very sure up to finding the sine of (A+B)
I'm pretty sure after that.
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