Solving Trigonometric Expressions: Finding the Value of AB

In this article, we will explore how to find the value of AB given that tan A frac{1}{2} and cot B 3. This involves the use of trigonometric identities and the tangent addition formula. Let's break down the steps and derive the solution.

Given that:- tan A frac{1}{2}- cot B 3 (Which implies tan B frac{1}{3} because cot B frac{1}{tan B}).

Step 1: Find the Angles A and B Using Trigonometric Functions

To find the angles, we use the arctangent function.

A tan^{-1} left(frac{1}{2}right)

This gives the principal value of the angle A.

B tan^{-1} left(frac{1}{3}right)

This gives the principal value of the angle B.

Step 2: Use the Tangent Addition Formula

The tangent addition formula is given by:

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

Substituting the values of tan A and tan B into the formula:

tan(A B) frac{frac{1}{2} frac{1}{3}}{1 - frac{1}{2} cdot frac{1}{3}}

Let's simplify the numerator and the denominator separately.

Numerator:

frac{1}{2} frac{1}{3} frac{3}{6} frac{2}{6} frac{5}{6}

Denominator:

1 - frac{1}{2} cdot frac{1}{3} 1 - frac{1}{6} frac{5}{6}

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

tan(A B) frac{frac{5}{6}}{frac{5}{6}} 1

This implies:

tan(A B) 1

Step 3: Concluding the Value of A B

Since tan(A B) 1, the angles that satisfy this equation are:

A B tan^{-1}(1)

Thus:

A B frac{pi}{4} npi quad text{for any integer } n

In the principal range, this simplifies to:

A B frac{pi}{4}

Therefore, the principal value of A B is:

A B frac{pi}{4}

This can also be expressed in degrees as 45°.

Verifications and Simplifications

Another quick verification is given as follows:

If tan A frac{1}{2} and cot B 3, which means cot A 2, then if we take the product cot A cdot cot B 6end{span}

Therefore, cot B - cot A 3 - 2 1

This implies:

cot(A B) 1

Hence:

A B tan^{-1}(1)

Thus:

A B frac{pi}{4}

Conclusion

The value of A B is indeed:

boxed{A B frac{pi}{4}}

Expressing in degrees is:

A B 45°

This conclusion aligns with both the direct computation and the verification through trigonometric identities.