Problem 8 - Algebra 2

Question

Solve each trigonometric equation for \(0 \leq \theta < 2 \pi\) $$ \sin \left(\frac{\pi}{2}-\theta\right)=-\cos (-\theta) $$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\)

Step 1: Use Co-Function Identity

Use the Co-Function Identity to change the sine term into a cosine term. The co-function identity states that \(\sin(\frac{\pi}{2}-\theta) = \cos(\theta)\). Using this, the equation becomes \(\cos(\theta) = -\cos(-\theta)\)

Step 2: Use Even/Odd Properties of Trigonometric Functions

Recall that the cosine function is even. This means that \(\cos(-\theta) = \cos(\theta)\). So the equation simplifies to \(\cos(\theta) = - \cos(\theta)\)

Step 3: Solve for Theta

Since we have \(\cos(\theta) = - \cos(\theta)\), the only way for this equation to be true is for \(\cos(\theta)\) to be zero. Therefore, \(\theta\) can be any angle at which cosine of the angle is zero , which are \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

Key Concepts

Co-Function Identity

In trigonometry, co-function identities are essential because they help relate sine and cosine functions. One of the key co-function identities is:

• \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \)

This identity tells us that the sine of an angle complement (measured from \( \frac{\pi}{2} \)) is equal to the cosine of the angle itself. This relationship is pivotal when solving trigonometric equations because it allows us to interchange sine and cosine, collaborating to simplify equations.
For example, in the given problem, we use this identity to rewrite \( \sin\left(\frac{\pi}{2} - \theta\right) \) as \( \cos(\theta) \), transforming our original equation into something more manageable. This simplification is valuable when you need to solve equations analytically and reduces potential complexity.

Even/Odd Properties

Trigonometric functions possess certain symmetry properties known as even and odd properties. These properties describe how the functions behave regarding negative angles:

• Even Function: \( \cos(-\theta) = \cos(\theta) \)
• Odd Function: \( \sin(-\theta) = -\sin(\theta) \)

A function is even if it is symmetric about the y-axis, meaning its graph doesn't change when \( \theta \) is replaced with \(-\theta\). The cosine function is even, so replacing \( \cos(-\theta) \) with \( \cos(\theta) \) is legitimate, which directly simplifies our problem further.
By observing the equation \( \cos(\theta) = -\cos(-\theta) \), and knowing that cosine is even, we can equate it to \( \cos(\theta) = -\cos(\theta) \). Recognizing these properties greatly aids in solving trigonometric equations, streamlining the process and minimizing errors.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of angles that satisfy the equation within a given range, here \(0 \leq \theta < 2\pi\). These solutions often lie at specific key points on the unit circle.
Given \( \cos(\theta) = -\cos(\theta) \), the elegance lies in recognizing that the cosine function can only equal its negative counterpart when it equals zero. This is because there’s no x-value except zero such that a number equals its negative. Thus, to solve, we set \( \cos(\theta) = 0 \).
The cosine function is zero at two principal points within one rotation of the unit circle:

• \( \theta = \frac{\pi}{2} \)
• \( \theta = \frac{3\pi}{2} \)

These points are the solutions to the equation in the specified range. Recognizing how to use the symmetries and identities of trig functions can help you solve these equations efficiently.