Problem 7 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The solutions to the trigonometric equation \(\cos(\pi/2 - \theta) = \csc(\theta)\) in the range \(0 \leq \theta < 2 \pi\) are \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).

Step 1: Convert to Equivalent Reciprocal Form

Rewrite \(\csc \theta\) as \(1/\sin \theta\). So, the equation \(\cos \left(\frac{\pi}{2}-\theta\right)=\csc \theta\) becomes \(\cos \left(\frac{\pi}{2}-\theta\right) = 1/\sin \theta\)

Step 2: Convert cosine to sine

We know that \(\cos(\pi/2 - \theta) = \sin\theta\). This is due to the co-function identity of sine and cosine. So, substitute \(\cos(\pi/2 - \theta)\) with \(\sin\theta\) in the equation: \(\sin \theta = 1/\sin \theta\).

Step 3: Solve for \( \theta \)

To solve the equation \(\sin \theta = 1/\sin \theta\), we square both sides to get rid of the fraction. This gives the equation \(\sin^2\theta = 1\). The solutions for \(\sin^2\theta = 1\) are \(\sin\theta = 1\) or \(\sin\theta = -1\). Solving for \( \theta \) in the interval \(0 \leq \theta < 2 \pi\), we get \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).

Step 4: Verify the solutions

Substitute \(\theta = \frac{\pi}{2}\) and \(\ \theta = \frac{3\pi}{2}\) into the original equation to ensure that they satisfy the equation.

Key Concepts

Co-function Identities

Co-function identities are mathematical relationships between trigonometric functions. When two angles are complementary, meaning they add up to \( \pi/2 \), their corresponding trigonometric functions are equal. For example, cosine and sine are co-functions, meaning

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

This identity is useful when solving equations because it allows us to convert a cosine function into a sine function. By using this identity, you can find equivalent expressions that make solving trigonometric equations easier.
Understanding these identities helps in simplifying equations and finding solutions within specific intervals, especially when dealing with angles measured in radians.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of one, centered at the origin of the coordinate system. The unit circle allows us to easily visualize and understand the behavior of trigonometric functions.
Each point on the circle corresponds to an angle \( \theta \) and has coordinates \((\cos \theta, \sin \theta)\). From these points:

• Angles in standard position are measured from the positive x-axis.
• The full circle represents the interval \(0 \leq \theta < 2\pi \).

Using the unit circle aids in finding the values of sine, cosine, and tangent for special angles such as \(\pi/2\) and \(3\pi/2\), which are part of our solution. The unit circle gives a visual insight into why these specific angles solve the equation \( \sin \theta = 1 \) or \( \sin \theta = -1 \).

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide another layer of depth in trigonometry. Functions like cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent, respectively.
The relevant function here is the cosecant function, defined as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Understanding reciprocals helps in converting and simplifying complex equations. In solving the given equation, replacing \( \csc \theta \) with its reciprocal form helps transform the problem into a more familiar one.
This conversion is essential when comparing two different trigonometric expressions, making it easier to evaluate and solve within the given interval.