Problem 47 - Algebra 2

Question

Find the complete solution in radians of each equation. $$ \sin ^{2} \theta-1=\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The solutions of the equation are \(\theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4\)

Step 1 : Convert the Equation Using Trig Identity

Use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\). This can be rearranged to \(\sin^2\theta = 1- \cos^2\theta\). Substitute this into the original equation: \((1 - \cos^2\theta) - 1 = \cos^2\theta\)

Step 2 : Simplify the Equation

Simplify the equation by removing like terms and bringing similar terms to one side of the equation. This leads to \(2\cos^2\theta = 1\)

Step 3 : Solve for \( \cos \theta \)

Solve the equation from step 2 for \( \cos \theta \). This results in \(\cos\theta = \sqrt{1/2}\) or \(\cos\theta = -\sqrt{1/2}\)

Step 4 : Find the Solutions

Recall the values of cosine from the unit circle. The solutions where \(\cos\theta = \sqrt{1/2}\) are \(\theta = \pi/4\) and \(\theta = 7\pi/4\). The solutions where \(\cos\theta = -\sqrt{1/2}\) are \(\theta = 3\pi/4\) and \(\theta = 5\pi/4\)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, serving as the backbone for many equations and transformations. It expresses the intrinsic relationship between the sine and cosine functions: \[ \sin^2\theta + \cos^2\theta = 1 \] This identity allows us to interchange trigonometric expressions and explore deeper algebraic manipulations. Based on the right triangle with hypotenuse 1, it reflects the unity aspect of trigonometric ratios. In this particular exercise, we utilize this identity to transform the given equation \( \sin^2\theta - 1 = \cos^2\theta \). By substituting terms using \( \sin^2\theta = 1- \cos^2\theta \), we simplify the equation's structure and generate an equation easily solvable for cosine values. Understanding and using the Pythagorean Identity is crucial for solving a broad range of trigonometric equations.

Unit Circle

The Unit Circle is a simple yet powerful tool in trigonometry representing the set of all points \((x, y)\) in the plane that are one unit away from the origin. Each point on this circle can be associated with an angle \(\theta\) measured in radians from the positive x-axis. This circle enables us to comprehend the angular positions and corresponding sine and cosine values clearly.

• On the unit circle, the point \((\cos\theta, \sin\theta)\) defines the coordinates for any angle \(\theta\).
• For angles like \(\theta = \pi/4\) and \(\theta = 3\pi/4\), the unit circle showcases that the corresponding cosine values are precisely \( \pm \sqrt{1/2} \).

In this exercise context, the unit circle helps us relate the values of \( \cos\theta = \pm\sqrt{1/2} \) to specific angles, enabling the identification of solutions such as \( \pi/4 \), \( 7\pi/4 \), \( 3\pi/4 \), and \( 5\pi/4 \). Understanding the geometry of the unit circle is key to mastering trigonometric equations.

Trigonometric Solutions

Trigonometric Solutions involve finding the angle values that satisfy a given trigonometric equation. The process requires understanding identities, properties of angles, and periodicity of trig functions. From simplifying the equation \( 2\cos^2\theta = 1 \), we determine \( \cos\theta = \pm\sqrt{1/2} \). This leads us to identify exact angles using known values from trigonometric tables or the unit circle.

• The values where \( \cos\theta = \sqrt{1/2} \) correspond to angles \( \theta = \pi/4 \) and \( \theta = 7\pi/4 \).
• For \( \cos\theta = -\sqrt{1/2} \), the angles become \( \theta = 3\pi/4 \) and \( \theta = 5\pi/4 \).

It's crucial to remember the periodic nature of trigonometric functions, which means any solution \(\theta\) can have multiple coterminal angles by adding an integer multiple of \( 2\pi \). Grasping trigonometric solutions ensures proficiency in tackling both practical problems and theoretical analyses.