If tan (A + B) = √3 and tan (A - B) = \(\frac{1}{\sqrt 3}\); 0° < (A + B) < 90°; A > B, then the values of A and B are _______ respectively.
45° and 15°
We are given a problem involving trigonometric equations with angles A and B. We need to find the values of angles A and B based on the given information.
Understanding the Given Information
We are provided with two main pieces of information:
- The value of the tangent of the sum of the angles (A + B).
- The value of the tangent of the difference of the angles (A - B).
Specifically, the given equations are:
- \(\tan (A + B) = \sqrt{3}\)
- \(\tan (A - B) = \frac{1}{\sqrt 3}\)
We are also given conditions on the angles:
- \(0^\circ < (A + B) < 90^\circ\): This means the sum of the angles is an acute angle.
- \(A > B\): Angle A is greater than angle B.
Recalling Standard Tangent Values
To solve these equations, we need to recall the standard values of the tangent function for common angles in the range \(0^\circ\) to \(90^\circ\).
- We know that \(\tan (60^\circ) = \sqrt{3}\).
- We know that \(\tan (30^\circ) = \frac{1}{\sqrt 3}\).
Setting Up Equations for Angles A and B
Comparing the given equations with the standard values, we can find expressions for (A + B) and (A - B).
From \(\tan (A + B) = \sqrt{3}\), and given that \(0^\circ < (A + B) < 90^\circ\), we can conclude:
\(A + B = 60^\circ\) (Equation 1)
From \(\tan (A - B) = \frac{1}{\sqrt 3}\), we can conclude:
\(A - B = 30^\circ\) (Equation 2)
Now we have a system of two linear equations with two variables, A and B.
Solving for Angles A and B
We can solve this system of equations using the elimination method.
- Add Equation 1 and Equation 2:
\((A + B) + (A - B) = 60^\circ + 30^\circ\)
\(A + B + A - B = 90^\circ\)
\(2A = 90^\circ\)
\(A = \frac{90^\circ}{2}\)
\(A = 45^\circ\)
- Substitute the value of A (\(45^\circ\)) into Equation 1:
\(45^\circ + B = 60^\circ\)
\(B = 60^\circ - 45^\circ\)
\(B = 15^\circ\)
Verifying the Conditions
We found A = \(45^\circ\) and B = \(15^\circ\). Let's check if these values satisfy the given conditions:
- Is \(0^\circ < (A + B) < 90^\circ\)? \(A + B = 45^\circ + 15^\circ = 60^\circ\). Yes, \(0^\circ < 60^\circ < 90^\circ\).
- Is \(A > B\)? \(45^\circ > 15^\circ\). Yes.
Both conditions are satisfied. The values A = \(45^\circ\) and B = \(15^\circ\) are correct.
Conclusion
The values of A and B are \(45^\circ\) and \(15^\circ\) respectively.
| Step | Calculation / Reasoning | Result |
|---|---|---|
| 1 | From \(\tan (A + B) = \sqrt{3}\) and \(0^\circ < (A + B) < 90^\circ\) | \(A + B = 60^\circ\) (Eq 1) |
| 2 | From \(\tan (A - B) = \frac{1}{\sqrt 3}\) | \(A - B = 30^\circ\) (Eq 2) |
| 3 | Add Eq 1 and Eq 2 | \(2A = 90^\circ\) |
| 4 | Solve for A | \(A = 45^\circ\) |
| 5 | Substitute A = \(45^\circ\) into Eq 1 | \(45^\circ + B = 60^\circ\) |
| 6 | Solve for B | \(B = 15^\circ\) |
| 7 | Verify conditions \(0^\circ < (A + B) < 90^\circ\) and \(A > B\) | \(60^\circ\) is in range, \(45^\circ > 15^\circ\). Conditions met. |
Revision Table: Trigonometric Ratios and Angles
Reviewing the tangent values for standard angles is crucial for solving such problems.
| Angle (\(\theta\)) | \(\tan(\theta)\) |
|---|---|
| \(0^\circ\) | 0 |
| \(30^\circ\) | \(\frac{1}{\sqrt 3}\) |
| \(45^\circ\) | 1 |
| \(60^\circ\) | \(\sqrt{3}\) |
| \(90^\circ\) | Undefined |
Additional Information: Solving Systems of Equations
The problem reduces to solving a system of two linear equations:
\(A + B = 60\)
\(A - B = 30\)
There are several methods to solve such systems, including substitution and elimination. In this solution, we used the elimination method by adding the two equations to eliminate B and solve for A. Then we substituted the value of A back into one of the original equations to solve for B.
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