Problem 112 - Precalculus with Limits

Question

In Exercises 109 - 112, use a graphing utility to solve the equation for \( \theta \), where \( 0 \le \theta < 2\pi \). \( \csc \theta = \sqrt{1 + \cot^2 \theta} \)

Step-by-Step Solution

Answer

The solutions to the equation are \( \theta = \pi/2, 3\pi/2 \)

Step 1: Convert to sine and cosine

First, convert the equation from cotangent and cosecant to cosine and sine, so it becomes easy for plotting. For this we can use the identities: \( \csc \theta = 1/\sin \theta \) and \( \cot \theta = \cos \theta/\sin \theta \). The equation becomes \(1/\sin \theta = \sqrt{1 + (\cos \theta/\sin \theta)^2}\)

Step 2: Simplify the equation

Next, simplify the equation. This can be done by squaring both sides of the equation (to eliminate the square root) and then multiplying each term by \(\sin^2 \theta\) to get rid of the fractions. This will give the equation in a more simple form of \( \sin^2 \theta = 1 + \cos^2 \theta\) which rearranges to \( \sin^2 \theta - \cos^2 \theta = 1 \).

Step 3: Express in the form of known trigonometric identity

Considering the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), substitute this in above equation. This gives \( 1 - \cos^2 \theta - \cos^2 \theta = 1 \), which simplifies to \( 2\cos^2 \theta = 0 \) and finally to \( \cos \theta = 0 \).

Step 4: Solve for theta

Now solve the equation \( \cos \theta = 0 \) for values of \(\theta\) within the prescribed range. You will have \( \theta = \pi/2, \ 3\pi/2 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying equations involving trigonometric functions. They allow us to transform complex expressions into simpler forms that are easier to work with. For example, in this exercise, we utilize two important identities:

• The identity for cosecant: \( \csc \theta = \frac{1}{\sin \theta} \)
• The identity for cotangent: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

By applying these identities, the original equation \( \csc \theta = \sqrt{1 + \cot^2 \theta} \) converts into an equation involving only sine and cosine. This process not only facilitates easier calculation but also demonstrates how identities streamline solving techniques. Recognizing and efficiently applying such identities are fundamental to mastering trigonometric equations.

Solving Equations Using Graphs

Solving trigonometric equations using graphs is a visual method that can illuminate the solutions clearly. When graphed, trigonometric functions reveal their periodic nature and the points where they intersect. In this exercise, where we solve \( \cos \theta = 0 \) within the interval \( 0 \leq \theta < 2\pi \), graphing provides a handy approach.When plotting \( y = \cos \theta \), the function impacts the horizontal axis at \( \theta = \pi/2 \) and \( \theta = 3\pi/2 \). These intercepts represent the solutions to our equation, showcasing how graphical methods verify analytical findings. This graphical approach complements algebraic methods, offering a double-checking mechanism.By utilizing a graphing tool or calculator, one can easily view these relationships and determine exact or approximate solutions, reinforcing comprehension of trigonometric equation behaviors.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that depict periodic oscillations. Each function represents specific relationships in a right-angled triangle:

• **Sine**: Represents the ratio of the length of the opposite side to the hypotenuse. It's given by \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• **Cosine**: Represents the ratio of the length of the adjacent side to the hypotenuse. It's calculated as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Both functions cycle every \(2\pi\), echoing at intervals of \(\pi\) and \(\pi/2\) respectively. They are interlinked through the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which is crucial in transforming expressions. In this problem, redefining the expression \( \sin^2 \theta - \cos^2 \theta = 1 \) into known identities helps in simplifying and solving equations, illustrating the flexible synergy of these functions.

Cosecant and Cotangent Identities

Cosecant and cotangent are reciprocal trigonometric functions that extend the use of sine and cosine.

• **Cosecant**: Reciprocal of sine, defined as \( \csc \theta = \frac{1}{\sin \theta} \).
• **Cotangent**: Reciprocal of tangent, defined as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These identities are particularly useful when dealing with trigonometric equations containing multiple functions. In our exercise's context, recognizing that \( \csc \theta = \sqrt{1 + \cot^2 \theta} \) connects these functions through the Pythagorean identity. Simplifying the equation to eliminate complex terms often requires utilizing these secondary trigonometric identities, thus emphasizing the interconnected nature of all trigonometric functions. Understanding these identities aids in transforming and solving otherwise intimidating equations, making problem-solving approachable.