Solving for tan A – B Using Trigonometric Identities

In this article, we are going to solve for ( tan A - B ) given certain conditions on angles A and B. We will use basic trigonometric identities and the tangent subtraction formula to find the value of ( tan A - B ).

Given Conditions and Calculations

Given:

Angle A is obtuse, and ( sin A frac{5}{13} ) Angle B is acute, and ( tan B frac{3}{4} )

Objective: To find the value of ( tan A - B ).

Step 1: Finding ( tan A )

Since angle A is obtuse, we first need to find ( cos A ) using the Pythagorean identity:

( sin^2 A cos^2 A 1 ) ( left(frac{5}{13}right)^2 cos^2 A 1 ) ( frac{25}{169} cos^2 A 1 ) ( cos^2 A 1 - frac{25}{169} ) ( cos^2 A frac{144}{169} )

Since A is obtuse, ( cos A ) will be negative:

( cos A -frac{12}{13} )

Now, we can find ( tan A ):

( tan A frac{sin A}{cos A} ) ( tan A frac{frac{5}{13}}{-frac{12}{13}} -frac{5}{12} )

Step 2: Finding ( tan B )

Since angle B is acute and the tangent of B is given:
( tan B frac{3}{4} )

Step 3: Applying the Tangent Subtraction Formula

We use the formula:

( tan (A - B) frac{tan A - tan B}{1 tan A tan B} )

Substituting the values:

( tan (A - B) frac{-frac{5}{12} - frac{3}{4}}{1 left(-frac{5}{12}right)left(frac{3}{4}right)} )

Step 4: Simplifying the Numerator and Denominator

Numerator:

( -frac{5}{12} - frac{3}{4} -frac{5}{12} - frac{9}{12} -frac{14}{12} -frac{7}{6} )

Denominator:

( 1 left(-frac{5}{12}right)left(frac{3}{4}right) 1 - frac{15}{48} frac{48}{48} - frac{15}{48} frac{33}{48} )

Step 5: Final Calculation

( tan (A - B) frac{-frac{7}{6}}{frac{33}{48}} -frac{7}{6} cdot frac{48}{33} -frac{56}{33} )

Therefore, the value of ( tan A - B ) is:

( boxed{-frac{56}{33}} )

Recall that ( tan A ) is negative because angle A is obtuse.

Conclusion

This article demonstrates how to solve for ( tan A - B ) using trigonometric identities. By applying the tangent subtraction formula and simplifying the expressions, we can find the exact value of ( tan A - B ).