Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank to complete the trigonometric identity. $$\sin ^{2} u+ \text{_____} =1$$

Step-by-Step Solution

Answer

The missing term is \(\cos^2 u\).

Step 1: Identify the trigonometric identity

Recognize the form of Pythagorean identity in trigonometry that is given by \(\sin^2(x) + \cos^2(x) = 1\). Here, \(u\) is used instead of x.

Step 2: Fill in the blank

Fill the blank with the term missing from the Pythagorean identity, which is \(\cos^2 u\).

Key Concepts

Secant Function

The secant function, denoted as \(\sec\), is one of the six fundamental trigonometric functions. It is closely related to the cosine function. The secant of an angle \(u\) is defined as the reciprocal of the cosine of that angle. In mathematical terms, the secant function is expressed as:
\[\sec(u) = \frac{1}{\cos(u)}\]The secant function may not be as commonly used as sine or cosine, but it plays an important role in various trigonometric identities and applications. It is primarily used when working with hyperbolas and in calculus during the integration of trigonometric expressions.

• Secant is undefined for values of \(u\) where \(\cos(u) = 0\), as division by zero is undefined.
  
• The secant function is periodic, with a period of \(2\pi\).

Negative Angle Identities

Trigonometric identities for negative angles are useful when simplifying expressions and solving trigonometric equations. These identities help transform functions of negative angles into equivalent positive angle expressions. For the secant function, the identity for a negative angle \(-u\) is reported as:
\[\sec(-u) = \sec(u)\]This identity stems from the even nature of the secant function, meaning that its graph is symmetric with respect to the y-axis. As a result:

• \(\sec(-u) = \frac{1}{\cos(-u)}\), and since \(\cos(-u) = \cos(u)\), it follows that \(\sec(-u) = \sec(u)\).
• The property holds for other even trigonometric functions like cosine.

By understanding negative angle identities, you can simplify and calculate trigonometric problems more efficiently. These identities are particularly critical in calculus, physics, and engineering.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that shows the relationship between sine and cosine functions. It is named after the famous Pythagorean theorem due to its similar form. The identity is expressed as:
\[\sin^2(u) + \cos^2(u) = 1\]This identity states that if you square the sine and cosine of the same angle and add them together, the result is always 1. The Pythagorean identity is pivotal in simplifying trigonometric expressions and solving equations, especially when dealing with angles and rotations.

• It can be rearranged to express either \(\sin^2(u)\) or \(\cos^2(u)\) in terms of the other:
  • \(\sin^2(u) = 1 - \cos^2(u)\)
  • \(\cos^2(u) = 1 - \sin^2(u)\)
• Application of this identity is essential when verifying other trigonometric identities.

Understanding the Pythagorean identity is crucial for higher-level mathematics where conversion and simplifications of trigonometric functions are common.