Problem 58 - Precalculus

Question

For the following exercises, simplify the given expression. $$ \cot \left(\frac{\pi}{2}-x\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \tan(x) \).

Step 1: Recall Trigonometric Identity

The expression \( \cot(\frac{\pi}{2}-x) \) is related to a known trigonometric identity for co-functions. The "co-function" identities state that \( \cot(\frac{\pi}{2} - x) = \tan(x) \). This comes from the fact that the cotangent of an angle is the same as the tangent of its complementary angle.

Step 2: Apply the Identity

Using the trigonometric identity from Step 1, substitute \( \tan(x) \) for \( \cot(\frac{\pi}{2}-x) \). Thus, the expression simplifies to \( \tan(x) \).

Key Concepts

Cotangent

The cotangent function, typically denoted as \( \cot(x) \), is one of the key trigonometric functions. It is defined as the reciprocal of the tangent function. In mathematical terms, this is represented as:

• \( \cot(x) = \frac{1}{\tan(x)} \)

Another way to understand cotangent is in relation to the sine and cosine functions:

• \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

Understanding these relationships helps in solving trigonometric problems, especially when simplification is required. It is important to remember that cotangent is undefined when the sine of the angle is zero because division by zero is not possible.

Tangent

Tangent, symbolized as \( \tan(x) \), is a fundamental trigonometric function. It is often understood as the ratio of the opposite side to the adjacent side of a right triangle. However, in terms of sine and cosine, it is expressed as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

Tangent is important in solving various problems involving angles, as it connects directly with other trigonometric functions through identities. Recognizing the formulas for tangent can assist in simplifying expressions, just like converting cotangent into tangent using identities. Additionally, it helps understand changes in function behavior when angles are manipulated.

Co-Function Identities

Co-function identities involve relationships between pairs of trigonometric functions whose arguments sum up to \( \frac{\pi}{2} \) (or 90 degrees). These identities include:

• \( \sin(\frac{\pi}{2} - x) = \cos(x) \)
• \( \cos(\frac{\pi}{2} - x) = \sin(x) \)
• \( \tan(\frac{\pi}{2} - x) = \cot(x) \)
• \( \cot(\frac{\pi}{2} - x) = \tan(x) \)

These identities are helpful because they allow you to replace one trig function with another when the angle is complementary. Recognizing these relationships is essential for simplifying expressions like \( \cot(\frac{\pi}{2} - x) \). Understanding co-function identities also provides deeper insights into the symmetry and periodicity of trigonometric functions.

Angle Simplification

Angle simplification is a technique used to transform trigonometric expressions into more manageable forms. By using identities like co-function and complementary angle identities, you can often reduce the complexity of expressions. For instance, \( \cot(\frac{\pi}{2} - x) \) simplifies directly to \( \tan(x) \) using the co-function identities.
This simplification is especially useful in calculation, making it less cumbersome to integrate or differentiate trigonometric expressions later. Mastering angle simplification enhances your ability to navigate various trigonometric problems with efficiency. This method is all about making smarter choices in expression handling, which saves time and reduces errors during both simple and complex trigonometric calculations.