Problem 18 - Algebra 2 and Trigonometry

Question

Prove that \(\tan \left(180^{\circ}+\theta\right)=\tan \theta\)

Step-by-Step Solution

Answer

\(\tan(180^{\circ} + \theta) = \tan \theta\) is true because \(\tan 180^{\circ} = 0\) and the tangent addition formula simplifies correctly.

Step 1: Understand the Identity

The problem asks us to prove the identity \(\tan(180^{\circ} + \theta) = \tan \theta\). This involves understanding how the tangent function behaves with respect to angle sum, particularly when adding \(180^{\circ}\).

Step 2: Recall the Tangent Addition Formula

The tangent addition formula states \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Here, we wish to apply this formula with \(a = 180^{\circ}\) and \(b = \theta\).

Step 3: Evaluate Tan(180°)

Recall that \(\tan 180^{\circ} = 0\), because the sine of \(180^{\circ}\) is \(0\) and the cosine is \(-1\).

Step 4: Apply the Tangent Addition Formula

Substitute \(a = 180^{\circ}\) and \(b = \theta\) into the tangent addition formula:\[tan(180^{\circ} + \theta) = \frac{\tan 180^{\circ} + \tan \theta}{1 - \tan 180^{\circ} \tan \theta}\]Since \(\tan 180^{\circ} = 0\), the expression simplifies to:\[tan(180^{\circ} + \theta) = \frac{0 + \tan \theta}{1 - 0 \cdot \tan \theta} = \tan \theta\]

Step 5: Conclude the Proof

Thus, we have proven that \(\tan(180^{\circ} + \theta) = \tan \theta\), confirming the given identity.

Key Concepts

Tangent Addition Formula

The tangent addition formula is a vital tool in trigonometry for combining angles within functions. It illustrates how the tangent of the sum of two angles can be expressed using the tangents of the individual angles. This formula is given by:\[tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]When using the formula, you break down the tangent of a sum into more manageable components. In our original exercise, this formula was the starting point for proving that \(\tan(180^{\circ} + \theta) = \tan \theta\). By substituting specific values, such as \(a = 180^{\circ}\) and \(b = \theta\), the formula can simplify complex expressions, making it easier to explore how tangents behave with combined angles. Remember, understanding how to apply this formula is crucial for exploring other trigonometric properties and identities.

Angle Sum Identities

Angle sum identities are essential tools that simplify the process of dealing with trigonometric functions that involve the addition of angles. They include identities for sine, cosine, and tangent functions. For the tangent function, the identity helps simplify the tangent of the sum of two angles and is expressed as:\[tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]Within the exercise, we used this identity to address the combination of a right angle (\(180^{\circ}\)) with another angle \(\theta\). By recognizing these identities, you can handle various trigonometric scenarios efficiently. Such identities are powerful when simplifying complex trigonometric expressions, making calculations more manageable, and uncovering deeper relationships between angles. These tools are foundational in studying and mastering trigonometry.

Properties of Tangent Function

To tackle trigonometric problems effectively, understanding the properties of the tangent function is crucial. Unlike sine and cosine, the tangent function extends to infinity, repeating every \(180^{\circ}\), which translates to a periodicity. This property is foundational in the original proof of \(\tan(180^{\circ} + \theta) = \tan \theta\). We see that tangent, being an odd function, mirrors through the origin:- \(\tan(-\theta) = -\tan \theta\)- Periodicity: \(\tan(\theta + 180^{\circ}) = \tan \theta\)In contexts involving additions or subtractions of \(180^{\circ}\), it directly implies repeating tangent values. Thus, knowing how the function behaves with angle additions can help unravel complex expressions or solve trigonometric equations. Mastery of these properties is beneficial not only for academic problems but also for real-world applications involving angles.