Problem 45 - Precalculus: Functions and Graphs

Question

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\sec \theta, \sin \theta$$

Step-by-Step Solution

Answer

\(\sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}}\)

Step 1: Recall Fundamental Trigonometric Identities

Start with the identity relating secant and cosine: \[ \sec \theta = \frac{1}{\cos \theta} \]Additionally, recall the Pythagorean identity that connects sine and cosine:\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Step 2: Express \(\cos \theta\) in Terms of \(\sin \theta\)

From the Pythagorean identity:\[ \sin^2 \theta + \cos^2 \theta = 1 \]Solving for \(\cos \theta\), we get:\[ \cos^2 \theta = 1 - \sin^2 \theta \]Thus,\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \](assuming \(\theta\) is acute and \(\cos \theta\) is positive).

Step 3: Express \(\sec \theta\) in Terms of \(\sin \theta\)

Substitute \(\cos \theta\) from Step 2 into the secant formula:\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\sqrt{1 - \sin^2 \theta}} \]This shows \(\sec \theta\) in terms of \(\sin \theta\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that connects the square of the sine and cosine functions of an angle. This identity states that for any angle \( \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \] This equation is reminiscent of the Pythagorean theorem. It forms the backbone of many trigonometric computations and transformations. By knowing this identity, you can express one trigonometric function in terms of another. For example, if you have the value of \( \sin \theta \), you can determine \( \cos \theta \) by rearranging the equation: \[ \cos^2 \theta = 1 - \sin^2 \theta \] This identity holds true for all angles \( \theta \), making it an indispensable tool for solving trigonometric equations and simplifying expressions.

Secant Function

The Secant Function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. It is defined as: \[ \sec \theta = \frac{1}{\cos \theta} \] The secant function shows how the cosine and secant are directly related by an inverse operation. When the angle \( \theta \) is such that \( \cos \theta \) is zero, the secant is undefined since division by zero is not possible.

• \( \sec \theta \) gets larger as \( \cos \theta \) gets closer to zero.
• For angles in the first quadrant where the angle \( \theta \) is acute, \( \sec \theta \) is positive.

In the context of expressing trigonometric functions in terms of each other, the secant function proves useful when you encounter transformations or need to simplify expressions involving cosine.

Sine Function

The sine function, represented as \( \sin \theta \), is a fundamental trigonometric function. It relates an angle in a right-angled triangle to the ratio of the length of the opposite side over the hypotenuse. Mathematically, it is expressed as: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \] For acute angles, where \( 0 < \theta < 90^\circ \), the sine of the angle provides a straightforward measurement of this ratio. Sine values range between -1 and 1, and knowing these properties is helpful in various applications:

• It is an essential part of the Pythagorean Identity, \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Sine function helps solve triangles and model periodic phenomena.

Understanding how sine is interconnected with cosine and other functions allows for more advanced trigonometric problem-solving.

Cosine Function

Cosine function, denoted \( \cos \theta \), describes the relationship between an angle and the ratio of the adjacent side over the hypotenuse in a right-angled triangle. The mathematical expression stands as: \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] Like the sine function, cosine varies between -1 and 1, and it reflects key characteristics of the angle \( \theta \). Specifically, the cosine function plays a pivotal role in:

• Using the Pythagorean Identity to relate cosine and sine functions.
• Defining the secant function through its reciprocal relationship.
• Simplifying complex trigonometric expressions by offering alternative forms.

The cosine function is fundamental in computations involving wave patterns, circular motion, and more, making it an indispensable part of trigonometry.