Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the identity. $$1+\tan ^{2} \theta=\square$$

Step-by-Step Solution

Answer

The completed identity is \(1 + \tan^{2}\theta = \sec^{2}\theta\).

Step 1: Understanding the problem

First, look at the expression and identify it as a form of the Pythagorean identity for the tangent function.

Step 2: Recall the Trigonometric Identity

Recall the Pythagorean identity involving the tangent function. In general, it is \(1 + \tan^{2}\theta = \sec^{2}\theta\).

Step 3: Substitute the Identity

Now substitute \(\sec^{2}\theta\) into the square box in the original equation, resulting in a complete identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine, cosine, and other trigonometric functions. One of the most common Pythagorean identities involves the tangent and secant functions:

• The core identity is: \(1 + \tan^{2}\theta = \sec^{2}\theta\).
• It arises from dividing the basic identity, \(\sin^{2}\theta + \cos^{2}\theta = 1\), by \(\cos^{2}\theta\).
• This transformation leads us to the relationship: \(\tan^{2}\theta = \frac{\sin^{2}\theta}{\cos^{2}\theta}\) and \(\sec^{2}\theta = \frac{1}{\cos^{2}\theta}\).

Recognizing these transformations allows us to solve a variety of trigonometric problems by relating these functions in a standardized form. These identities serve as building blocks for solving complex equations and verifying other identities.

Tangent Function

The tangent function is one of the primary trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle.

• It is expressed as \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).
• The function is periodic with a period of \(\pi\) and is undefined where the cosine is zero, which results in vertical asymptotes.

Tangent is crucial for expressing angles and slopes and is used extensively in calculus and geometry.
The relationship \(1 + \tan^{2}\theta = \sec^{2}\theta\) highlights how tangent connects with secant, providing deeper insights into angle measures.

Secant Function

The secant function is less commonly used in comparison to sine and cosine but plays an important role in trigonometry.

• It is defined as the reciprocal of cosine: \(\sec\theta = \frac{1}{\cos\theta}\).
• This makes the secant function undefined where the cosine is zero.

Secant helps in various applications such as solving trigonometric equations and is useful in calculating lengths using its inverse relationships.
In the Pythagorean identity \(1 + \tan^{2}\theta = \sec^{2}\theta\), secant ties in with tangent to form a complete conceptual picture of these angles and ratios, aiding in solving trigonometric problems more efficiently.