Question

Solve \(\sin ^{2} x=\cos ^{2} x+1\) algebraically for all values of \(x .\) Give your answer(s) in radians.

Short answer

x = \frac{\pi}{2} + 2k\pi or \frac{3\pi}{2} + 2k\pi, where k is any integer.

Step-by-step solution

Recall the Pythagorean Identity

The Pythagorean identity states that \[\begin{equation} \sin^2 x + \cos^2 x = 1 \end{equation}\]

Substitute the Identity into the Equation

Rewrite the equation \sin^2 x = \cos^2 x + 1 using the Pythagorean identity: \[\begin{equation} \sin^2 x = 1 - \sin^2 x + 1 \end{equation}\]

Simplify the Equation

Combine like terms to simplify the equation: \[\begin{equation} \sin^2 x = 2 - \sin^2 x \end{equation}\]

Solve for \sin^2 x

Add \sin^2 x to both sides to get: \[\begin{equation} 2\sin^2 x = 2 \end{equation}\]. Then divide both sides by 2 to find: \[\begin{equation} \sin^2 x = 1 \end{equation}\]

Take the Square Root of Both Sides

Take the square root of both sides to solve for \sin x: \[\begin{equation} \sin x = \pm 1 \end{equation}\]

Determine the General Solution

Recall that \sin x = 1 at \[\begin{equation} x = \frac{\pi}{2} + 2k\pi \end{equation}\] and \sin x = -1 at \[\begin{equation} x = \frac{3\pi}{2} + 2k\pi \end{equation}\], where k is any integer.

Key concepts

Pythagorean identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that:\[ \sin^2 x + \cos^2 x = 1 \]This equation is true for any angle \(x\). The identity is derived from the Pythagorean theorem applied to a unit circle. In a unit circle, the radius is 1, and any point on the circle can be represented as \((\cos x, \sin x)\). Therefore, the sum of the squares of these coordinates is always 1.We use the Pythagorean identity to simplify trigonometric equations and prove other trigonometric identities. For instance, to solve the equation \(\sin^2 x = \cos^2 x + 1\), we substitute \(\cos^2 x\) with \(1 - \sin^2 x\) using the Pythagorean identity. This substitution is the first key step toward finding the solution.

solving trigonometric equations

Solving trigonometric equations involves finding all the angles that satisfy the given equation. To solve an equation like \(\sin^2 x = \cos^2 x + 1\), follow these steps:1. **Use known identities:** Start with known trigonometric identities. For instance, use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to rewrite the equation.2. **Substitute and simplify:** Substitute \(\cos^2 x\) with \(1 - \sin^2 x\) in the equation. Then simplify the resulting equation. \[ \sin^2 x = 1 - \sin^2 x + 1 \] Combine like terms: \[ 2 \sin^2 x = 2 \] Divide both sides by 2: \[ \sin^2 x = 1 \]3. **Solve for the variable:** Take the square root of both sides to find \(\sin x = \pm 1\).4. **Determine all solutions:** Recall the values of \(x\) where \(\sin x = 1\) and \(\sin x = -1\). These occur at specific points on the unit circle, typically written as general solutions to account for all possible angles.By following these steps, you ensure that all possible values of \(x\) are found.

general solution for sine function

The general solution for the sine function helps us find all angles \(x\) that satisfy a trigonometric equation involving \(\sin x\). If we know that \(\sin x = k\), where \(k\) is some constant, the general solutions are derived from analyzing the unit circle.For example:

• When \(\sin x = 1\), the solutions occur at \(x = \frac{\pi}{2} + 2k\pi\), where \(k\) is any integer.
• When \(\sin x = -1\), the solutions occur at \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is any integer.

These solutions include positive and negative angles that repeat every full rotation (\(2\pi\) radians). By expressing solutions this way, we capture all possible angles that meet the equation's criteria. For the given problem, we use these general forms to determine all the values of \(x\) that satisfy \(\sin x = \pm 1\). Understanding these general solutions is crucial for solving more complex trigonometric equations effectively.