Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Use the Pythagorean identities to write the expression as an integer. (a) \(7 \sec ^{2} \gamma-7 \tan ^{2} \gamma\) (b) \(7 \sec ^{2}(\gamma / 3)-7 \tan ^{2}(\gamma / 3)\)

Step-by-Step Solution

Answer

Both expressions simplify to 7.

Step 1: Identify the Pythagorean Identity

The Pythagorean identity related to secant and tangent is: \[ \sec^2 \theta - \tan^2 \theta = 1 \] This identity can be applied to simplify expressions that involve \( \sec \) and \( \tan \).

Step 2: Apply Identity to Expression (a)

For the expression: \[ 7 \sec^2 \gamma - 7 \tan^2 \gamma \]Factor out the 7 from both terms:\[ 7(\sec^2 \gamma - \tan^2 \gamma) \]Use the identity \( \sec^2 \gamma - \tan^2 \gamma = 1 \):\[ 7 \cdot 1 = 7 \]

Step 3: Apply Identity to Expression (b)

For the expression: \[ 7 \sec^2(\gamma / 3) - 7 \tan^2(\gamma / 3) \]Factor out the 7 from both terms:\[ 7(\sec^2(\gamma / 3) - \tan^2(\gamma / 3)) \]Use the identity where \( \theta = \gamma / 3 \), hence \( \sec^2(\gamma / 3) - \tan^2(\gamma / 3) = 1 \):\[ 7 \cdot 1 = 7 \]

Key Concepts

Secant Function

In trigonometry, the secant function is one of the six fundamental trigonometric functions. It is closely related to the cosine function. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine of that angle. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \). This means that wherever the cosine value is zero, the secant will become undefined, since division by zero is not possible.
Understanding the secant function is paramount when simplifying expressions that involve secant squared terms, especially when used with Pythagorean identities. These identities help transform complex expressions into simpler forms, which can be easily evaluated.
For instance, knowing that \( \sec^2 \theta = 1 + \tan^2 \theta \) is vital. This transformation is helpful in recognizing expressions like \( \sec^2 \theta - \tan^2 \theta \) as it equates to 1 based on the rearranged Pythagorean identity.

Tangent Function

The tangent function is another key element of trigonometry and plays an integral role in certain trigonometric identities such as the Pythagorean identities. The tangent of an angle \( \theta \) is defined as the ratio of the sine of the angle to the cosine of the angle: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
In the context of trigonometric identities, the tangent function often appears in expressions involving both \( \tan \) and \( \sec \) functions. Understanding these relationships can simplify your calculations significantly.
To simplify trigonometric expressions, you use the relationship \( \sec^2 \theta - \tan^2 \theta = 1 \). This can be employed to recognize and simplify terms, leading to reduced expressions like \( 7 \sec^2 \gamma - 7 \tan^2 \gamma \) into a constant integer, 7 in this case.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler, more manageable forms. The Pythagorean identities serve as powerful tools in this regard, as they provide a foundation for transforming expressions with \( \sec \) and \( \tan \) functions.
The identity \( \sec^2 \theta - \tan^2 \theta = 1 \) allows for direct simplification of expressions like \( 7(\sec^2 \gamma - \tan^2 \gamma) \). By recognizing this specific configuration, you can quickly substitute the identity and simplify to an integer.

• Factor out common terms: Look for common factors across terms that you can factor out, reducing the complexity of the expression.
• Apply identities: Use known identities smartly to replace complex parts of the expression with simpler equivalents.
• Simplify the result: After applying identities, further simplify the result to achieve a clear and concise answer.

Using these strategies makes trigonometry more digestible and is instrumental in solving complex equations with greater ease.