Problem 47 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall Identity for Sine and Cosine

The sine and cosine functions are related by the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity can be used to express \( \sin \theta \) in terms of \( \cos \theta \).

Step 2: Express Cosine Using Secant

The secant function is the reciprocal of the cosine function, so \( \sec \theta = \frac{1}{\cos \theta} \). Consequently, \( \cos \theta = \frac{1}{\sec \theta} \).

Step 3: Substitute Cosine in Pythagorean Identity

Replace \( \cos \theta \) in the Pythagorean identity with \( \frac{1}{\sec \theta} \): \[ \sin^2 \theta + \left( \frac{1}{\sec \theta} \right)^2 = 1 \]. This results in: \[ \sin^2 \theta + \frac{1}{\sec^2 \theta} = 1 \].

Step 4: Solve for Sine

Rearrange the equation from the previous step to solve for \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \frac{1}{\sec^2 \theta} \]. Take the square root of both sides to express \( \sin \theta \) in terms of \( \sec \theta \): \[ \sin \theta = \sqrt{1 - \frac{1}{\sec^2 \theta}} \].

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is denoted as \( \sin \theta \) where \( \theta \) is an angle. For right triangles, the sine of an angle is defined as the ratio between the length of the side opposite the angle to the length of the hypotenuse.
In mathematical terms, for a right triangle:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. Understanding sine is important as it helps calculate and understand wave patterns, circular motion, and oscillations, among many other applications.

Secant Function

The secant function, denoted as \( \sec \theta \), is a less frequently used trigonometric function compared to sine and cosine. However, it plays a crucial role in understanding reciprocal relationships. The secant function is defined as the reciprocal of the cosine function:

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

Secant is related to the hypotenuse and the adjacent side of a right triangle, but its primary importance lies in its reciprocal property.
This function is particularly useful in calculus and trigonometric integrals, where it provides another perspective to tackle problems involving cosine. Note that the secant function is undefined wherever the cosine function is zero, as division by zero is mathematically undefined.

Pythagorean Identity

The Pythagorean identity is a critical concept in trigonometry that relates the sine and cosine functions. It is expressed as:

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

This identity comes from the Pythagorean theorem applied in a unit circle. In a right triangle, this relationship helps in expressing one trigonometric function in terms of another. The identity is extremely useful, particularly when you need to reframe equations involving trigonometric functions into simpler forms.
In this exercise, the identity is used to transform \( \sin \theta \) into an expression using the secant function, showcasing its power in simplifying trigonometric equations.

Acute Angles

An acute angle is one that measures less than \( 90^\circ \) or \( \frac{\pi}{2} \) radians. In trigonometry, acute angles are often considered within the context of right triangles where only angles less than \( 90^\circ \) occur (besides the right angle itself).
Understanding acute angles is essential because:

• They are part of calculations in trigonometric expressions.
• Most basic trigonometric functions assume the context of acute angles initially.

By focusing on acute angles, students can simplify problems and ensure calculations remain straightforward, avoiding complexities that arise with angle measures larger than \( 90^\circ \). In exercises like this, acknowledging angles as acute ensures the assumptions and identities applied remain valid and easy to manage.