Problem 13 - Algebra and Trigonometry

Question

\(3-16 \cdot\) Solve the given equation. $$ \sin \theta-1=\cos \theta $$

Step-by-Step Solution

Answer

Solutions: \( \theta = n\pi \) and \( \theta = \frac{\pi}{2} + m\pi \) where \( n, m \) are integers.

Step 1: Rearrange the Equation

Start by rearranging the given equation to bring like terms on one side. The given equation is: \( \sin \theta - 1 = \cos \theta \). Move \( \cos \theta \) to the left side by subtracting \( \cos \theta \) from both sides:\[ \sin \theta - \cos \theta = 1 \]

Step 2: Square Both Sides

To simplify the trigonometric expression, square both sides of the rearranged equation:\[ (\sin \theta - \cos \theta)^2 = 1^2 \]Expanding the left side gives:\[ \sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = 1 \]

Step 3: Apply Pythagorean Identity

Use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substituting into the equation, replace \( \sin^2 \theta + \cos^2 \theta \) with 1:\[ 1 - 2\sin \theta \cos \theta = 1 \]This simplifies to:\[ -2\sin \theta \cos \theta = 0 \]

Step 4: Solve for Trigonometric Terms

Divide both sides by -2 to isolate the trigonometric terms:\[ \sin \theta \cos \theta = 0 \]This implies that either \( \sin \theta = 0 \) or \( \cos \theta = 0 \).

Step 5: Find General Solutions

Solve the resulting equations separately:1. \( \sin \theta = 0 \) implies \( \theta = n\pi \) where \( n \) is an integer.2. \( \cos \theta = 0 \) implies \( \theta = \frac{\pi}{2} + m\pi \) where \( m \) is an integer.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone relationships in trigonometry. It's named after the Pythagorean theorem because it relates the squares of the sides of a right triangle. The identity itself is simple: \[\sin^2 \theta + \cos^2 \theta = 1\]This means that for any angle \(\theta\), the square of its sine plus the square of its cosine will always equal one. This can be visualized on the unit circle, where the radius is one and any terminal side of angle \(\theta\) corresponds to a point on the circle with coordinates \((\cos \theta, \sin \theta)\). Thus, the formula confirms that these coordinates always lie on the circle. Applying the Pythagorean Identity simplifies problems by allowing terms to cancel or be expressed in alternative forms. In trigonometric equations like the one we are solving, substituting \(\sin^2 \theta + \cos^2 \theta\) by 1 often makes the equation easier to manage. This step was fundamental in moving from a complex trigonometric equation to a simpler one, aiding in our solution.

General Solutions

Finding the general solutions to a trigonometric equation means identifying all possible angles \(\theta\) that satisfy the equation. Trigonometric functions like sine and cosine are periodic, meaning they repeat their values in regular intervals. Thus, there are often infinitely many solutions to these equations. For the equation \(\sin \theta = 0\), the solutions are values of \(\theta\) that make the sine function zero. This happens at angles that are integer multiples of \(\pi\): \[\theta = n\pi, \, \text{where} \, n \text{ is an integer.}\]Similarly, for \(\cos \theta = 0\), the solutions occur at intervals of \(\frac{\pi}{2}\) from the starting point:\[\theta = \frac{\pi}{2} + m\pi, \, \text{where} \, m \text{ is an integer.}\]These general solutions provide all the angles that satisfy the initial equation, allowing one to understand the full scope of possible solutions across all rotations. It's crucial to account for these periodic solutions, especially when tackling trigonometric problems.

Sine and Cosine Relations

Sine and cosine functions are two of the most basic trigonometric functions, representing ratios derived from right triangles. They have interesting and useful characteristics:

• Periodicity: Both sine and cosine functions have a period of \(2\pi\), meaning their values repeat every \(2\pi\) radians.
• Symmetry: Sine is an odd function, following \(\sin(-\theta) = -\sin(\theta)\), whereas cosine is even, with \(\cos(-\theta) = \cos(\theta)\).
• Range: Both functions have values ranging from -1 to 1.

In solving trigonometric equations, understanding these properties helps simplify and predict the behavior of these functions across different equations. For instance, if two angles differ by a multiple of the period, their sine and cosine will be identical. Moreover, these relations underpin the behavior of the Pythagorean Identity, establishing the foundation for broader applications in solving trigonometric equations.