Solving Trigonometric Equations to Find Angles A and B

We are given two trigonometric equations involving angles A and B, along with certain conditions. Our goal is to find the values of angles A and B using these equations and conditions.

The given equations are:

1. \(tan(A-B)=\dfrac{1}{\sqrt3}\)
2. \(tan(A+B)=\sqrt 3\)

We are also given the conditions:

• \(0^{\circ}<\angle(A+B)<90^{\circ}\)
• \(\angle A > \angle B\)

Step 1: Using the inverse tangent function

We know the standard values of the tangent function for specific angles. Using the first equation:

\(tan(A-B)=\dfrac{1}{\sqrt3}\)

We know that \(tan(30^{\circ}) = \dfrac{1}{\sqrt3}\). Therefore, we can write:

\(A-B = 30^{\circ}\) (Equation 1)

Now, using the second equation:

\(tan(A+B)=\sqrt 3\)

We know that \(tan(60^{\circ}) = \sqrt 3\). Therefore, we can write:

\(A+B = 60^{\circ}\) (Equation 2)

This value \(A+B = 60^{\circ}\) satisfies the condition \(0^{\circ}<\angle(A+B)<90^{\circ}\).

Step 2: Solving the system of linear equations

We now have a system of two linear equations with two variables, A and B:

• \(A - B = 30^{\circ}\)
• \(A + B = 60^{\circ}\)

We can solve this system by adding the two equations together:

\((A - B) + (A + B) = 30^{\circ} + 60^{\circ}\)

\(2A = 90^{\circ}\)

Now, divide by 2 to find the value of A:

\(A = \dfrac{90^{\circ}}{2}\)

\(A = 45^{\circ}\)

Step 3: Finding the value of Angle B

Substitute the value of A (45°) into either of the original equations. Let's use Equation 2:

\(A + B = 60^{\circ}\)

\(45^{\circ} + B = 60^{\circ}\)

Subtract 45° from both sides to find B:

\(B = 60^{\circ} - 45^{\circ}\)

\(B = 15^{\circ}\)

Step 4: Verifying the conditions

We found \(A = 45^{\circ}\) and \(B = 15^{\circ}\).

Let's check the condition \(\angle A > \angle B\):

\(45^{\circ} > 15^{\circ}\)

This condition is satisfied.

Let's check the condition \(0^{\circ}<\angle(A+B)<90^{\circ}\):

\(A+B = 45^{\circ} + 15^{\circ} = 60^{\circ}\)

\(0^{\circ} < 60^{\circ} < 90^{\circ}\)

This condition is also satisfied.

Conclusion

The values of angles A and B that satisfy the given trigonometric equations and conditions are \(A = 45^{\circ}\) and \(B = 15^{\circ}\).

Revision Table: Standard Tangent Values

It's helpful to remember the tangent values for some standard angles:

Additional Information: Solving Systems of Equations

A system of linear equations like the one we solved (\(A - B = 30^{\circ}\) and \(A + B = 60^{\circ}\)) can be solved using various methods, including:

• Substitution Method: Solve one equation for one variable (e.g., solve \(A-B=30^{\circ}\) for A to get \(A=B+30^{\circ}\)), and then substitute this expression into the other equation.
• Elimination Method: Add or subtract the equations (as we did) to eliminate one variable, leaving a single equation with one variable.

In this case, the elimination method by adding the equations was straightforward because the 'B' terms had opposite signs.