Question

In a triangle \(\mathrm{ABC}\), if \(\operatorname{atan} \mathrm{A}+\operatorname{btan} \mathrm{B}=(\mathrm{a}+\mathrm{b}) \tan \frac{\mathrm{A}+\mathrm{B}}{2}\) then (a) \(2 \mathrm{~A}=\mathrm{B}\) (b) \(\mathrm{A}=2 \mathrm{~B}\) (c) \(\mathrm{A}=3 \mathrm{~B}\) (d) \(\mathrm{A}=\mathrm{B}\)

Short answer

(a) \(A \gt B\) (b) \(A \lt B\) (c) \(B = A + C\) (d) \(A = B\) Solution: Based on the given formula and the trigonometric identities, we derived that \(\tan{A} = \tan{B}\), which implies that angles A and B are equal. Therefore, the correct answer is (d) \(A = B\).

Step-by-step solution

Recall basic triangle properties

Recall that in a triangle, the sum of the three angles is always \(180^{\circ}\). Therefore, in triangle ABC: \[ A + B + C = 180^{\circ} \]

Write the given formula in terms of tangent

The formula given for angle A and B is: \[a\tan{A} + b\tan{B} = (a+b) \tan{\frac{A+B}{2}}\]

Apply the tangent addition formula

Recall that the tangent addition formula states that \(\tan{(A + B)} = \frac{\tan{A} + \tan{B}}{1 - \tan{A}\tan{B}}\). Using this formula, we can write the formula in Step 2 as: \[a \tan{A} + b \tan{B} = (a+b) \frac{\tan{A} + \tan{B}}{1 - \tan{A}\tan{B}}\]

Solve for \(\tan{(A+B)}\)

Now, create a common denominator and continue simplification to get: \[a \tan{A}(1 - \tan{A}\tan{B}) + b \tan{B}(1 - \tan{A}\tan{B}) = (a+b) (\tan{A} + \tan{B}) \]

Simplify and solve for the relationship between A and B

Now expand and simplify the equation to get: \[(a \tan{A} - ab\tan{A} \tan{B}) + (b \tan{B} - ab \tan{A} \tan{B}) = (a+b) (\tan{A} + \tan{B})\] Notice that \(ab\tan{A}\tan{B}\) is a common term to both terms on the left hand side of the equation. So we can simplify the equation further: \[a \tan{A} + b \tan{B} = (a + b)(\tan{A} + \tan{B})\] Now divide both sides by \((a+b)\): \[\frac{a \tan{A} + b \tan{B}}{a+b} = \tan{A} + \tan{B}\] And continue to simplify: \[\frac{a \tan{A} + b \tan{B} - a\tan{A} - b\tan{B}}{a + b} = 0\] This simplifies to: \[\tan{A} = \tan{B}\] Since their tangents are equal, angles A and B must also be equal. So the correct answer is: (d) \(A = B\)

Key concepts

Triangle Properties

In any triangle, the properties and relationships between its angles and sides form the basis of solving many trigonometric problems. One fundamental property you should always remember is that the sum of the interior angles in any triangle is always 180 degrees. This means that for a triangle with angles \(A\), \(B\), and \(C\):

• \(A + B + C = 180^{\circ}\)

Understanding this property helps determine if a problem is dealing with a valid triangle. Additionally, knowing other properties, such as the Pythagorean theorem for right triangles, can be useful for identifying specific types of triangles or solving for missing values. These fundamental properties form the foundation for many trigonometric relationships used in solving triangle-based problems.

Tangent Addition Formula

The tangent addition formula is a key identity used in trigonometry when dealing with the sum of two angles. It is particularly useful in problems requiring the computation of the tangent of the sum of two angles, \( (A + B) \). The formula is given by:

• \( \tan{(A + B)} = \frac{\tan{A} + \tan{B}}{1 - \tan{A} \tan{B}} \)

This formula arises frequently in problems involving non-right triangles and their angles, as it helps bridge the connection between the tangent values of individual angles and their sums. To apply this, ensure you have or can calculate the tangent values for each individual angle in question. This formula is a handy tool for transforming problems into simpler equations that can be solved more easily.

Angle Relationships

When working with geometric shapes like triangles, understanding the relationships between angles can be extremely helpful. Such relationships guide us in solving complex trigonometric equations. In our case, if we know \( \tan{A} = \tan{B} \), we conclude that angles \( A \) and \( B \) are equal because tangent is a periodic function with a period of \( 180^{\circ} \), meaning it repeats every half turn.

• For a given range, \( \tan{A} = \tan{B} \) implies \( A = B + n \times 180^{\circ} \) (where \( n \) is an integer), but in triangle context, typically we'd consider their angles being equal for simplicity.

Grasping these concepts allows for more informed decisions when dealing with triangle problems, particularly with trigonometric identities involving angle measures.

Tangent Function

The tangent function is one of the primary trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle:

• \( \tan{\theta} = \frac{\text{Opposite}}{\text{Adjacent}} \)

It is important to note that tangent can have both positive and negative values depending on the quadrant in which the angle lies:

• Positive in the first and third quadrants
• Negative in the second and fourth quadrants

This understanding becomes crucial when applying identities like the tangent addition formula or when evaluating tangent values in specific angle relationships within triangles. In the context of triangles, especially non-right ones, using tangent and its properties helps solve for angular changes or sides, thus providing a clearer understanding of the problem at hand.