Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \tan \left(\theta+\frac{\pi}{2}\right)=-\cot \theta $$

Step-by-Step Solution

Answer

The identity is verified by understanding their periodic and complementary properties.

Step 1: Understanding the Problem

The problem asks us to verify the identity \( \tan \left( \theta + \frac{\pi}{2} \right) = -\cot \theta \). This means that we need to show that the tangent of \( \theta + 90^{\circ} \) is equal to the negative cotangent of \( \theta \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The identity is verified by understanding their periodic and complementary properties.

Key Concepts

Tangent Function: Basics and Properties

The tangent function is a fundamental trigonometric function, usually abbreviated as \( \tan \). It is defined as the ratio of the sine function to the cosine function: \[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

• It is an odd function, meaning \( \tan(-\theta) = -\tan \theta \).
• Tangent is periodic with a period of \( \pi \) radians, or 180 degrees. This means \( \tan(\theta + \pi) = \tan \theta \).

The tangent function is undefined when the cosine of \( \theta \) is zero. These occur at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. The graph of the tangent function features vertical asymptotes where the function is undefined. Understanding the basic properties of the tangent function helps in verifying identities and solving trigonometric equations. It’s frequently used in geometry, physics, and engineering for calculating angles and slopes.

Cotangent Function: Definition and Key Points

The cotangent function, denoted as \( \cot \), is closely related to the tangent function. It is defined as the reciprocal of the tangent function: \[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]

• Cotangent is also an odd function: \( \cot(-\theta) = -\cot \theta \).
• Similar to tangent, the cotangent function is periodic with a period of \( \pi \).

The cotangent function is undefined when the sine of \( \theta \) is zero. Specifically, it is undefined at \( \theta = n\pi \), where \( n \) is an integer. The graph of the cotangent function contains vertical asymptotes at these points. Cotangent is often useful for working with angles in triangle problems and for converting between different trigonometric functions.

Angle Addition Formulas

Angle addition formulas are essential tools in trigonometry. These formulas provide a way to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the functions of the angles themselves. For tangent, the addition formula is: \[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\] This formula is particularly useful for verifying and simplifying complex trigonometric expressions.

• For the example given, \( \tan \left(\theta + \frac{\pi}{2}\right) \), notice that \( \tan \frac{\pi}{2} \) is undefined. However, using the properties of the tangent function, we can transform and simplify.
• This setup leads to understanding why \( \tan \left(\theta + \frac{\pi}{2}\right) \) can be expressed as \( -\cot \theta \).
• Angle addition identities are fundamental for solving trigonometric equations and verifying identities like the original exercise.

Learning how to apply these formulas simplifies many trigonometric problems and expands your mathematical toolbox.