Solving the Trigonometric Expression Problem

The problem asks us to find the value of the expression: tan 27° tan 34° + tan 34° tan 29° + tan 29° tan 27°.

Let the three angles be \(A = 27^\circ\), \(B = 34^\circ\), and \(C = 29^\circ\). The expression we need to evaluate is \(\tan A \tan B + \tan B \tan C + \tan C \tan A\).

First, let's find the sum of these three angles:

• \(A + B + C = 27^\circ + 34^\circ + 29^\circ\)
• \(A + B + C = 61^\circ + 29^\circ\)
• \(A + B + C = 90^\circ\)

Since the sum of the angles is \(90^\circ\), we have \(A + B = 90^\circ - C\).

Using Trigonometric Identities for Angles Summing to 90°

We can use the tangent function with the relationship \(A + B = 90^\circ - C\). Taking the tangent of both sides:

\(\tan(A + B) = \tan(90^\circ - C)\)

We know two important trigonometric identities:

• The tangent addition formula: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
• The complementary angle identity: \(\tan(90^\circ - C) = \cot C = \frac{1}{\tan C}\) (provided \(\tan C \neq 0\))

Since the angles are 27°, 34°, and 29°, their tangents are non-zero and well-defined.

Equating the two expressions for \(\tan(A+B)\) and \(\tan(90^\circ - C)\):

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

Solving the Equation to Find the Value

Now, we can cross-multiply the equation:

\((\tan A + \tan B) \tan C = 1 - \tan A \tan B\)

Distribute \(\tan C\) on the left side:

\(\tan A \tan C + \tan B \tan C = 1 - \tan A \tan B\)

To get the form of the expression we need to evaluate, move the term \(\tan A \tan B\) from the right side to the left side:

\(\tan A \tan B + \tan A \tan C + \tan B \tan C = 1\)

Rearranging the terms in the order given in the question:

\(\tan A \tan B + \tan B \tan C + \tan C \tan A = 1\)

Substituting the original angles back:

\(\tan 27^\circ \tan 34^\circ + \tan 34^\circ \tan 29^\circ + \tan 29^\circ \tan 27^\circ = 1\)

Thus, the value of the given expression is 1.

Step-by-Step Evaluation

Here is a summary of the steps:

1. Identify the angles: \(A = 27^\circ, B = 34^\circ, C = 29^\circ\).
2. Calculate the sum of the angles: \(A + B + C = 27^\circ + 34^\circ + 29^\circ = 90^\circ\).
3. Recognize that \(A + B = 90^\circ - C\).
4. Take the tangent of both sides: \(\tan(A + B) = \tan(90^\circ - C)\).
5. Apply the tangent addition formula and the complementary angle identity: \(\frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{1}{\tan C}\).
6. Cross-multiply: \((\tan A + \tan B) \tan C = 1 - \tan A \tan B\).
7. Expand the left side: \(\tan A \tan C + \tan B \tan C = 1 - \tan A \tan B\).
8. Rearrange the terms to get the required expression: \(\tan A \tan B + \tan B \tan C + \tan C \tan A = 1\).

The final value of the expression is 1.

Revision Table: Key Concepts

Additional Information: General Identity

This problem illustrates a general trigonometric identity. If \(A, B, C\) are angles such that \(A + B + C = 90^\circ\), then it is always true that:

\(\tan A \tan B + \tan B \tan C + \tan C \tan A = 1\)

This identity is derived using the same steps as shown in the solution. It is a useful result to remember for similar problems involving tangent of angles that sum up to \(90^\circ\).

Another related identity for angles summing to \(180^\circ\) is also important:

If \(A + B + C = 180^\circ\), then \(\tan A + \tan B + \tan C = \tan A \tan B \tan C\).

Understanding these identities helps in solving various trigonometry problems efficiently.