Let sin A = 12/13 with 90º≤A≤180º and tan B = -4/3 with 270º≤B≤360º. Find tan (A + B).
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To find the value of tan(A + B), we first need to find the values of A and B.
Given that sin A = 12/13 and 90º ≤ A ≤ 180º, we can find A by using the inverse sin function (also called arcsin or sin^-1) on a calculator.
sin^-1(12/13) = 67.38º
So, A = 67.38º.
Similarly, given that tan B = -4/3 and 270º ≤ B ≤ 360º, we can find B by using the inverse tan function (also called arctan or tan^-1) on a calculator.
tan^-1(-4/3) = -53.13º
However, this angle (-53.13º) is in the fourth quadrant, which is between 270º and 360º. We need to add 360º to this angle to bring it into the desired range between 0º and 90º.
-53.13º + 360º = 306.87º
So, B = 306.87º.
Now that we have the values of A and B, we can find tan(A + B) by using the formula for the sum of angles in trigonometry:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
Plugging in the values:
tan(A + B) = (tan 67.38º + tan 306.87º) / (1 - tan 67.38º * tan 306.87º)
Since tan 306.87º is the same as tan (-53.13º) (because they are reference angles), we can rewrite the formula as:
tan(A + B) = (tan 67.38º + tan (-53.13º)) / (1 - tan 67.38º * tan (-53.13º))
Now, we can use a calculator to find the individual values of tan 67.38º and tan (-53.13º) and substitute them into the formula to calculate the final answer for tan(A + B).
Note: Some calculators may require using parentheses for the negative angles when finding the tangent.