Problem 59 - Precalculus

Question

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

Step-by-Step Solution

Answer

\(\tan \left( \frac{\pi}{2} - x \right) = \cot x\).

Step 1: Understand the Identity

The expression \( \tan \left(\frac{\pi}{2} - x\right) \) can be simplified using a trigonometric identity. This expression is related to the co-function identity for tangent.

Step 2: Apply the Co-Function Identity

According to the co-function identity \( \tan \left( \frac{\pi}{2} - x \right) = \cot x \). This leverages the fact that the tangent of an angle is equal to the cotangent of its complement.

Step 3: Simplify the Expression

Recognizing the identity, the expression \( \tan \left( \frac{\pi}{2} - x \right) \) simplifies directly to \( \cot x \).

Key Concepts

Co-Function Identity

The co-function identity is a crucial concept in trigonometry. It's a direct way to relate trigonometric functions that are complements of each other. A complementary angle is two angles that add up to 90 degrees (or \( \frac{\pi}{2} \) radians). For tangent, the co-function identity states that \( \tan\left(\frac{\pi}{2} - x\right) = \cot x \). This means that the tangent of the complement of an angle is the same as the cotangent of the angle itself. These identities help transform one trigonometric expression into another. They simplify equations and make calculations easier.

• These identities are derived from the sine and cosine functions since tangent and cotangent are ratios involving these functions.
• They are especially handy in integration or differentiation scenarios where transforming the function can simplify the process.

Understanding these identities expands your ability to work with trigonometric expressions efficiently.

Tangent

The tangent function is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. In terms of the unit circle, tangent relates to sine and cosine as \[\tan x = \frac{\sin x}{\cos x}.\]When working with angles,

• The tangent of an angle tells you the slope of the line if the angle is placed in standard position.
• Tangent periods repeat every \( \pi \) radians, unlike sine and cosine which have periods of \( 2\pi \).

Understanding tangent's behavior, especially its asymptotes where cosine is zero, is critical. Also, tangent reaches very large values approaching these asymptotes, leading to vertical leaps in its graph.Exploring how tangent interacts with its complementary function, cotangent, through identities like the co-function identity helps deepen the understanding of both functions.

Cotangent

Cotangent is another fundamental trigonometric function. It's the reciprocal of the tangent function. That means\[\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}.\]Cotangent offers a perspective that's complementary to tangent. Since it's a reciprocal, it helps in understanding inverse behaviors in sine and cosine effectively.

• The primary characteristic of cotangent is its alternatively located asymptotes and zeros compared to tangent.
• These occur at different intervals, which can couple with tangent in nice symmetry, forming the classic wave patterns.

Using cotangent efficiently in trigonometric identities allows flexible simplification and expression of equations. This is particularly evident when applying the co-function identity to switch between tangent and cotangent, streamlining complex trigonometric calculations.