Problem 29 - Algebra and Trigonometry

Question

Use an identity to find the value of each expression. Do not use a calculator. $$ \sec ^{2} 23^{\circ}-\tan ^{2} 23^{\circ} $$

Step-by-Step Solution

Answer

The value of the expression \( \sec^2 23^{\circ} - \tan^2 23^{\circ} \) is 1.

Step 1: Identify the required trigonometric identity

The provided equation includes secant squared and tangent squared. For these, the Pythagorean identity is applicable, which is \( \sec^2 \theta - \tan^2 \theta = 1 \).

Step 2: Apply the identity

We apply this identity to the given equation and substitute \(\theta\) by 23 degrees. So our equation becomes: \( \sec^2 23^{\circ} - \tan^2 23^{\circ} \). According to the identity, this simplifies to 1.

Step 3: Provide the Answer

Following the application of the trigonometric identity, we obtain our answer as 1.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the foundational identities in trigonometry. It connects the square of the secant function and the tangent function in a simple formula. This identity is derived from the similar relations among sine, cosine, and their squares. The standard form of this identity is given by the equation:\[\sec^2 \theta = \tan^2 \theta + 1\]This equation can also be rewritten as:\[\sec^2 \theta - \tan^2 \theta = 1\]This transformation is incredibly useful when simplifying trigonometric expressions. It shows that for any angle \( \theta \), the difference of squares of secant and tangent yields a constant value of 1. The identity holds true for all angles where these functions are defined, making it a powerful tool in trigonometry. Keep it in mind whenever you're working with secant and tangent squares, as it can drastically simplify your calculations!

Secant Function

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. In other words:\[\sec \theta = \frac{1}{\cos \theta}\]This function represents the ratio of the hypotenuse to the adjacent side in a right triangle. Because it involves division by the cosine of an angle, the secant function is undefined wherever the cosine equals zero, such as at \( \theta = 90^\circ \) or \( \theta = 270^\circ \).

• The range of the secant function is \((-\infty, -1]\text{ or }[1, \infty)\).
• It is periodic with a period of \(360^\circ \) or \(2\pi\) radians.
• Secant shares properties with cosine, but its graph includes vertical asymptotes at the points where cosine is zero.

Understanding secant is crucial when working with trigonometric identities, especially those involving Pythagorean identities. Always remember its reciprocal relationship with cosine to solve various trigonometric equations effectively.

Tangent Function

The tangent function, represented as \( \tan \theta \), is one of the basic trigonometric functions. It is defined as the ratio between the sine and cosine functions:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]Alternatively, in a right-angled triangle, tangent represents the ratio of the opposite side to the adjacent side.

• The tangent function is undefined when the cosine of angle \( \theta \) is zero, which generally occurs at \( \theta = 90^\circ \), \(270^\circ\), etc.
• It has a period of \(180^\circ \) or \(\pi\) radians, making its graph present repeating patterns across the coordinate plane.
• The function's range includes all real numbers, \((-\infty, \infty)\).

The tangent function is essential in various trigonometric identities, particularly in the Pythagorean identity, where it pairs with the secant function to simplify complex expressions. Remembering its relationship with sine and cosine is crucial for solving many trigonometric problems.