Problem 10 - Algebra 2

Question

Solve each trigonometric equation for \(0 \leq \theta < 2 \pi\) $$ \sin ^{2} \theta+\cos ^{2} \theta=\sin \theta $$

Step-by-Step Solution

Answer

\(\theta = \frac{ \pi}{2}\)

Step 1: Identify the standard identities

Identify that the left side of the equation resembles the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\). Thus, we can rewrite the equation to \(1 = \sin \theta\).

Step 2: Solve for \(\theta\)

After rewriting, solve for \(\theta\) by finding the arcsine of 1 which yields \(\theta = \frac{ \pi}{2}\).

Step 3: Check the solution

Finally, validate whether the solution is within the given interval [0, 2π). In this case, \(\frac{ \pi}{2}\) is within this range, hence it is accepted as the solution.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that for any angle \( \theta \), the sum of the squares of sine and cosine equals one: \[\sin^{2} \theta + \cos^{2} \theta = 1\] This identity is derived from the Pythagorean theorem and applies to all angles. It's a useful tool for simplifying complex trigonometric equations. You can recognize it by identifying when the expression \( \sin^{2} \theta + \cos^{2} \theta \) appears, allowing us to substitute with 1 where suitable. In solving the given trigonometric equation, we substituted the left side of the equation with 1, as per this identity, leading us to: \[1 = \sin \theta\] This step was pivotal in simplifying and subsequently solving the equation effectively.

Arcsine Function

The arcsine function, denoted as \( \arcsin \), is the inverse of the sine function. It is used to find an angle when the value of its sine is known. For example, when we solve \( 1 = \sin \theta \), we use the arcsine function to determine \( \theta \).

• \( \arcsin(1) = \frac{\pi}{2} \)

This tells us that \( \theta = \frac{\pi}{2} \), since the sine of \( \frac{\pi}{2} \) is 1. The range of the arcsine function stems between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), making it essential to ensure solutions fall within this interval when dealing with inverse trigonometric problems. The arcsine function provides a direct way to calculate angles with known sine values, important for solving trigonometric equations.

Interval Notation

Interval notation is a way of representing subsets of real numbers, often used to specify the domain or range of a function. For example, in this problem, we are asked to find \( \theta \) in the interval \([0, 2\pi) \).

• The square bracket \([\)] indicates inclusion, meaning 0 is part of the solution.
• The parenthesis \(()\) indicates exclusion, meaning \(2\pi\) is not a part of the solution.

Thus, the interval \([0, 2\pi)\) includes all \( \theta \) values starting from 0 and stopping just before \( 2\pi \). It is crucial in trigonometry to ensure your solutions fall within the specified interval since trigonometric functions are periodic and can have multiple solutions. After solving the equation, confirming that \( \theta = \frac{\pi}{2} \) falls within this interval ensures the solution is valid for the problem constraints.