Problem 59 - Algebra and Trigonometry

Question

Use the first Pythagorean identity to prove the second. [Hint: Divide by \(\cos ^{2} \theta . ]\)

Step-by-Step Solution

Answer

The second Pythagorean identity is \(\tan^2 \theta + 1 = \sec^2 \theta\).

Step 1: Recall the First Pythagorean Identity

The first Pythagorean identity is given by the equation:\[sin^2 \theta + \cos^2 \theta = 1\]This is a fundamental identity in trigonometry, connecting the squares of sine and cosine functions to 1.

Step 2: Normalize by \(\cos^2 \theta\)

To derive the second Pythagorean identity, divide each term of the first identity by \(\cos^2 \theta\):\[\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}\]

Step 3: Simplify the Fractions

Each term in the equation can be simplified. The second term simplifies to 1 and the first term uses the definition of tangent. The equation becomes:\[\tan^2 \theta + 1 = \sec^2 \theta\]This is the second Pythagorean identity, expressing the relationship between tangent and secant functions.

Key Concepts

Pythagorean identity

The Pythagorean identities are among the most fundamental relationships in trigonometry, providing connections between the sine, cosine, tangent, and secant functions. The first Pythagorean identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity derives its name from the Pythagorean Theorem, as it reflects the relationship of squares akin to those used in a right triangle. It states that when you're dealing with a right triangle, the square of the sine of an angle plus the square of the cosine of that same angle will always equal one.
The first identity serves as a starting point to derive other useful identities in trigonometry. Dividing this base identity by \( \cos^2 \theta \) helps us connect to the second identity, bringing in tangent and secant functions. Ultimately, these identities assist in simplifying more complex trigonometric expressions and solving trigonometric equations.
Understanding these identities is crucial for navigating through trigonometry, as they present basic truths about angles and their relationships in a circled coordinate system.

tangent function

The tangent function is an essential trigonometric function that arises from the sine and cosine functions. In a right triangle, tangent relates an angle to the ratio of the length of the opposite side to the length of the adjacent side. Analytically, tangent can be defined as:

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

Incorporating the tangent function into the Pythagorean identity, as shown in the original exercise, helps us uncover new trigonometric relationships. When dividing the first Pythagorean identity by \( \cos^2 \theta \), the term \( \frac{\sin^2 \theta}{\cos^2 \theta} \) simplifies to \( \tan^2 \theta \). This simplification highlights how tangent is connected to both sine and cosine, enabling the formulation of further identities.
Tangent is particularly useful in solving various trigonometric problems and can help model periodic phenomena, including waves and oscillations, making it valuable in fields like physics and engineering.

secant function

The secant function is another fundamental trigonometric function closely connected to the cosine. In the context of right triangles, secant is defined as the ratio of the hypotenuse to the adjacent side. Mathematically, secant is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In the context of Pythagorean identities, secant becomes essential when deriving the second Pythagorean identity. By dividing each term of the first identity \( \sin^2 \theta + \cos^2 \theta = 1 \) by \( \cos^2 \theta \), we've already seen that the term \( \frac{1}{\cos^2 \theta} \) appears, which simplifies to \( \sec^2 \theta \).
The emergence of the secant function in this identity emphasizes its role in expressing relationships among the trigonometric functions. Secant functions are frequently used in solving equations where only cosine is involved but a reciprocal representation simplifies the process. Understanding how secant interplays with other trig functions broadens the ability to handle complex trigonometric evaluations, especially when dealing with functions involving reciprocals.