Understanding the Trigonometry Problem

The problem gives us two pieces of information about two angles, \(\alpha\) and \(\beta\). We are told that their sum is \(\frac{\pi}{4}\) radians (which is equal to 45 degrees) and that \(2 \tan \alpha = 1\). Our goal is to find the value of \(\tan 2\beta\).

To solve this, we will first find the value of \(\tan \alpha\) from the second given equation. Then, we will use the first equation relating \(\alpha\) and \(\beta\) to find \(\tan \beta\). Finally, we will use the double angle formula for tangent to find \(\tan 2\beta\).

Step-by-Step Solution for tan 2β

Step 1: Find the value of tan α

We are given the equation:

\(2 \tan \alpha = 1\)

To find \(\tan \alpha\), we just need to divide both sides by 2:

\(\tan \alpha = \frac{1}{2}\)

So, we know the value of \(\tan \alpha\).

Step 2: Use the sum of angles to find tan β

We are given that:

\(\alpha + \beta = \frac{\pi}{4}\)

We can take the tangent of both sides of this equation:

\(\tan(\alpha + \beta) = \tan(\frac{\pi}{4})\)

We know that \(\tan(\frac{\pi}{4}) = 1\). So, the equation becomes:

\(\tan(\alpha + \beta) = 1\)

Now, we use the tangent addition formula, which is \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). Applying this formula for \(\tan(\alpha + \beta)\):

\(\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} = 1\)

We already found that \(\tan \alpha = \frac{1}{2}\). Substitute this value into the equation:

\(\frac{\frac{1}{2} + \tan \beta}{1 - (\frac{1}{2}) \tan \beta} = 1\)

To solve for \(\tan \beta\), multiply both sides by the denominator \((1 - \frac{1}{2} \tan \beta)\):

\(\frac{1}{2} + \tan \beta = 1 \times (1 - \frac{1}{2} \tan \beta)\)

\(\frac{1}{2} + \tan \beta = 1 - \frac{1}{2} \tan \beta\)

Now, gather the terms involving \(\tan \beta\) on one side and the constant terms on the other side:

\(\tan \beta + \frac{1}{2} \tan \beta = 1 - \frac{1}{2}\)

Combine the terms:

\((1 + \frac{1}{2}) \tan \beta = \frac{1}{2}\)

\(\frac{3}{2} \tan \beta = \frac{1}{2}\)

To isolate \(\tan \beta\), multiply both sides by \(\frac{2}{3}\):

\(\tan \beta = \frac{1}{2} \times \frac{2}{3}\)

\(\tan \beta = \frac{1}{3}\)

So, we have found the value of \(\tan \beta\).

Step 3: Calculate tan 2β using the double angle formula

We need to find the value of \(\tan 2\beta\). We can use the tangent double angle formula, which is \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). In our case, \(\theta\) is \(\beta\).

So, \(\tan 2\beta = \frac{2 \tan \beta}{1 - \tan^2 \beta}\)

We found that \(\tan \beta = \frac{1}{3}\). Substitute this value into the formula:

\(\tan 2\beta = \frac{2 \times (\frac{1}{3})}{1 - (\frac{1}{3})^2}\)

First, calculate the numerator and the term in the denominator:

• Numerator: \(2 \times \frac{1}{3} = \frac{2}{3}\)
• Denominator term: \((\frac{1}{3})^2 = \frac{1^2}{3^2} = \frac{1}{9}\)

Now substitute these back into the formula for \(\tan 2\beta\):

\(\tan 2\beta = \frac{\frac{2}{3}}{1 - \frac{1}{9}}\)

Calculate the denominator:

\(1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{9 - 1}{9} = \frac{8}{9}\)

So, the expression for \(\tan 2\beta\) becomes:

\(\tan 2\beta = \frac{\frac{2}{3}}{\frac{8}{9}}\)

To divide fractions, we multiply the numerator by the reciprocal of the denominator:

\(\tan 2\beta = \frac{2}{3} \times \frac{9}{8}\)

Multiply the numerators and denominators:

\(\tan 2\beta = \frac{2 \times 9}{3 \times 8} = \frac{18}{24}\)

Finally, simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 6:

\(\tan 2\beta = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}\)

Final Answer for tan 2β

Based on the calculations, the value of \(\tan 2\beta\) is \(\frac{3}{4}\).

Revision Table: Key Trigonometry Steps

Additional Information on Trigonometric Identities

This problem heavily relies on trigonometric identities, specifically the tangent addition formula and the tangent double angle formula. Understanding these formulas is crucial for solving problems involving sums and multiples of angles.

• Tangent Addition Formula: This formula relates the tangent of the sum of two angles to the tangents of the individual angles.

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

  A similar formula exists for the difference of angles:

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

• Tangent Double Angle Formula: This is a special case of the addition formula where \(A = B = \theta\).

  \(\tan(2\theta) = \tan(\theta + \theta) = \frac{\tan \theta + \tan \theta}{1 - \tan \theta \tan \theta} = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)

  This formula allows us to find the tangent of double an angle if we know the tangent of the angle itself.

Mastering these identities helps in simplifying expressions and solving various trigonometric equations.