Problem 125 - A Graphical Approach to Precalculus with Limits

Question

Use fundamental identities to find each expression. Write \(\cos \theta\) in terms of \(\sin \theta\) if \(\theta\) is acute.

Step-by-Step Solution

Answer

\( \cos \theta = \sqrt{1 - \sin^2 \theta} \) for acute \( \theta \).

Step 1: Recall Pythagorean Identity

The Pythagorean identity is one of the most fundamental trigonometric identities:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity relates the square of the sine and cosine functions of the same angle.

Step 2: Rearrange for \( \cos^2 \theta \)

Since we want \( \cos \theta \) in terms of \( \sin \theta \), we need \( \cos^2 \theta \). Rearrange the Pythagorean identity to solve for \( \cos^2 \theta \):\[ \cos^2 \theta = 1 - \sin^2 \theta \]

Step 3: Take the Square Root

To find \( \cos \theta \), we take the square root of both sides of the equation:\[ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \]Since \( \theta \) is an acute angle, \( \cos \theta \) is positive.

Step 4: Write the Final Expression

Since \( \cos \theta \) must be positive for acute angles, our expression becomes:\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]This is the expression for \( \cos \theta \) in terms of \( \sin \theta \) for an acute angle.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone of trigonometry, encapsulating a fundamental relationship between sine and cosine functions. It's named after the Pythagorean Theorem because of its geometric foundation in right triangles. The identity states that the sum of the squares of sine and cosine of an angle equals one. Mathematically, it is represented as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity holds true for all angles and serves as the basis for many trigonometric transformations and simplifications.

• It helps in expressing one trigonometric function in terms of another, providing a pathway for solving trigonometric equations.
• Given any angle \( \theta \), if you know either \( \sin \theta \) or \( \cos \theta \), you can find the other using this identity.

Whether solving equations or verifying identities, the Pythagorean Identity is an indispensable tool that provides mathematical elegance and simplicity.

Sine and Cosine Relationship

The relationship between sine and cosine functions lies at the heart of trigonometry. These two functions describe the fundamental oscillatory behavior of angles. With respect to a unit circle, sine and cosine functions are defined as the coordinates of a point rotating around the circle:

• \( \sin \theta \) represents the y-coordinate.
• \( \cos \theta \) represents the x-coordinate.

This relationship is key when converting between different trigonometric expressions. An interesting aspect of their relationship is the ability to express one function in terms of the other using identities like the Pythagorean Identity:\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]Such transformations are particularly useful in scenarios where an angle’s specific function is needed due to missing or unknown values. The exploration of the sine and cosine relationship allows for flexibility and efficiency in trigonometric calculations.

Acute Angle Properties

Acute angles, defined as angles less than 90 degrees, possess distinct properties that make trigonometric calculations straightforward. One important property of acute angles is that both sine and cosine values are positive.

• For \( 0 < \theta < 90^{\circ} \), \( \sin \theta > 0 \) and \( \cos \theta > 0 \).
• This positivity is crucial when evaluating expressions and identities where only the principal (positive) value is required.

In the context of our solution, since \( \theta \) is an acute angle, it simplifies the result of \( \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \) to \( \cos \theta = \sqrt{1 - \sin^2 \theta} \), without the ambiguity of sign.Acute angles are prevalent in many real-world applications, from engineering to architecture, where precise angle measurements and calculations are vital.