Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \tan (\pi-\theta)=-\tan \theta $$

Step-by-Step Solution

Answer

The reduction formula \( \tan(\pi-\theta)=-\tan\theta \) is verified.

Step 1: Understand the Trigonometric Identity

The formula we need to verify is \( \tan(\pi - \theta) = -\tan\theta \). This suggests a relationship between the tangent of an angle and the tangent of its supplementary angle. The task is to prove that the tangent of an angle subtracted from \( \pi \) is the negative of the tangent of the original angle.

Step 2: Use Tangent Angle Subtraction Formula

Recall the tangent subtraction formula: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \). We'll use this formula with \( A = \pi \) and \( B = \theta \) to find \( \tan(\pi - \theta) \).

Step 3: Calculate \( \tan(\pi) \)

The value of \( \tan(\pi) \) is 0 because tangent is sine over cosine, and \( \sin(\pi) = 0 \), while \( \cos(\pi) = -1 \). So, \( \tan(\pi) = \frac{0}{-1} = 0 \).

Step 4: Apply the Subtraction Formula

Substitute \( A = \pi \) and \( B = \theta \) into the tangent subtraction formula:\[\tan(\pi - \theta) = \frac{\tan(\pi) - \tan(\theta)}{1 + \tan(\pi) \tan(\theta)}\]Replacing \( \tan(\pi) \) with 0, the formula simplifies to:\[\tan(\pi - \theta) = \frac{0 - \tan(\theta)}{1 + 0 \cdot \tan(\theta)} = \frac{-\tan(\theta)}{1} = -\tan(\theta)\]

Step 5: Conclusion

Through the calculation, we verified that \( \tan(\pi - \theta) \) equals \(-\tan\theta\). This confirms the given reduction formula is true for any angle \( \theta \).

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry and relates the ratio of two specific distances in a right-angled triangle: the length of the opposite side to the length of the adjacent side. The formula is given by:

• For an angle \( \theta \), the tangent function is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• It can also be viewed in terms of the sine and cosine functions, as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Tangent values have periodic properties, repeating every \( \pi \) radians (180 degrees), and exhibit symmetry around the origin. This periodicity means that the tangent of an angle \( \theta \) will be the same as the tangent of the angle plus any integer multiple of \( \pi \). These properties make the tangent function very useful in analyzing angles and their respective lines, especially in calculus and physics. It's essential to remember that the tangent is undefined whenever the cosine of \( \theta \) is zero, leading to vertical asymptotes in its graph.

Angle Subtraction Formula

The angle subtraction formula is an identity used to find the tangent of a difference of two angles. This is particularly useful when angles cannot be neatly expressed as sums or simple fractions. The formula is:

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

When working through problems, you substitute the angles for \( A \) and \( B \) into this formula. This allows for the evaluation of the tangent of more complex angle expressions. For instance, evaluating \( \tan(\pi - \theta) \) involves recognizing that \( A = \pi \) and substituting into the formula:

• Thus it simplifies to \( \tan(A - B) = \frac{0 - \tan(\theta)}{1 + 0 \cdot \tan(\theta)} = -\tan(\theta) \)

Understanding this step demonstrates how trigonometric identities can simplify expressions and provide insights into angle relationships.

Supplementary Angles

Supplementary angles are pairs of angles that add up to \( \pi \) radians or 180 degrees. In the context of trigonometric identities, these can reveal interesting symmetries and properties. For example, if you have an angle \( \theta \), its supplementary angle is \( \pi - \theta \).

• For tangent, this leads to identity: \( \tan(\pi - \theta) = -\tan(\theta) \).

This means that the tangent of a supplementary angle is the negative of the tangent of the original angle. This identity stems from how trigonometric functions handle reflections and symmetries. The subtraction of \( \pi \) essentially reflects the angle across the y-axis on the unit circle, altering the sign of sine, while cosine remains unchanged, resulting in the negative tangent value.Recognizing these patterns not only simplifies complex trigonometric calculations but also deepens an understanding of the geometric interpretation of trigonometric functions.