if tanA=(5/12) and A is in quadrant 3, and cosB=(-3/5) is in quadrant 2, what is the value of sin(A− B)?

Answer 1

sin (A - B) = sin A cos B - cos A sin B
sin (A - B) = (- 5/13) (- 3/5) - (- 12/13) (4/5)
sin (A - B) = 15 / 65 + 48 / 65
sin (A - B) = 63 / 65

Answer 2

In quadrant 3 tan is positive. I could rewrite 5/12 into -5/-12.
First, draw angle A in standard position and label it "A". You can then create a triangle in the third quadrant. Since tan is opposite over adjacent, -5 will be the side opposite of the reference angle, while -12 will be the adjacent side. Lable those points on the triangle. Next we can easily find the hypotenuse by using pythagorus.

-5^2 + (-12)^2 = r^2
169 = r^2
13 = r

This will come in handy later on.

Similarily, you should do the same thing with angle B, using a different graph though. Draw angle B in standard position. Label it "B". The adjacent side of the reference angle will be -3 and hypotenuse will be 5. Find the other side by using pythagorus.

-3^2 + y^2 = 5^2
9 + y^2 = 25
y^2 = 16
y = 4

By using the trig identities:
sin(A− B)
= sin A cos B - cos A sin B

We only know one thing, cos B. So, that means we must find sin A, cos A, and sin B.

Look back at the first triangle. Sin is opposite over hypotenuse, so, -5/13 = sin A.

To get cos A, adjacent over hypotenuse. -12/13 = cos A

For sin B, we must use the second triangle. Opposite over hypotenuse, therefore 4/5 = sin B

Finally we can plug in those numbers into the equation.

sin(A− B)
= sin A cos B - cos A sin B
= (-5/13)(-3/5) - (-12/13 )(4/5)
= (3/13) - (48/65)
= (15/ 65) - (48/65)
= -33/65

Answer 3

For angle A... you use the arctan... or the inverse tan...
A = arctan(5/12) = 22.61986495
However, since A is in quadrant 3, you add 180 to the answer... and you get...
A = 202.62

For angle B, use inverse cosine...
B = arccos(-3/5) = 126.8698976
Since this angles is already in quadrant 2, you can just leave it as it is...

so, A - B would be... 75.7499673
sin(A-B) is .9692

hope that helps...

Answer 4

tan A = (5/12)
arctan (5/12) = 22.62 degrees (below the negative x-axis)

cos B = (-3/5)
arccos B = 126.87 degrees (counterclockwise from positive x-axis)

A-B = 22.62 - 126.87 = -104.51 degrees
sin (-104.51) = -0.968

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