Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta < 360^{\circ}\) that satisfy each equation. $$ \cot ^{2} \theta=\csc \theta+1 $$

Step-by-Step Solution

Answer

\( \theta = 48.6^{\circ} \) and \( \theta = 131.4^{\circ} \).

Step 1: Understand the Equation

The given equation is \( \cot^2 \theta = \csc \theta + 1 \). Our aim is to find values of \( \theta \) in the interval \( 0^{\circ} \leq \theta < 360^{\circ} \).

Step 2: Use Trigonometric Identities

Recall the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Substitute these into the equation: \( \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{1}{\sin \theta} + 1 \).

Step 3: Simplify the Equation

Simplify the left side: \( \frac{\cos^2 \theta}{\sin^2 \theta} \). The right side becomes \( \frac{1 + \sin \theta}{\sin \theta} \). Equate the numerators and denominators: \( \cos^2 \theta = 1 + \sin \theta \cdot \sin^2 \theta \).

Step 4: Transform into a Quadratic Equation

Use \( \cos^2 \theta = 1 - \sin^2 \theta \), leading to \( 1 - \sin^2 \theta = 1 + \sin \theta - \sin^2 \theta \). Cancel terms and rearrange to form a quadratic: \( \sin \theta (-1 - 1) = - \sin \theta \).

Step 5: Solve for \( \sin\theta \)

The quadratic becomes \( \sin^2 \theta + \sin \theta - 1 = 0 \). Solve using the quadratic formula: \( \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a=1, b=1, c=-1 \).

Step 6: Simplify Solutions from Quadratic Formula

Calculate the discriminant: \( 1^2 - 4 \times 1 \times (-1) = 5 \). Then \( \sin \theta = \frac{-1 \pm \sqrt{5}}{2} \). Only \( \sin \theta = \frac{-1 + \sqrt{5}}{2} \) is in the range \([-1, 1]\).

Step 7: Find Values of \( \theta \)

Compute \( \theta \) using \( \sin^{-1} \left(\frac{\sqrt{5}-1}{2}\right) \). The angles due to periodicity are approximately \( 48.6^{\circ} \) and \( 180^{\circ} - 48.6^{\circ} = 131.4^{\circ} \) because \( \sin \theta = \sin (180^{\circ} - \theta) \).

Step 8: Verify the Solutions

Substitute \( \theta = 48.6^{\circ} \) and \( 131.4^{\circ} \) back into the original equation to verify they satisfy \( \cot^2 \theta = \csc \theta + 1 \).

Key Concepts

Cotangent

The cotangent function, commonly abbreviated as "cot," is one of the fundamental trigonometric functions. It is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. Mathematically, it is expressed as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This function is the reciprocal of tangent and is undefined when \( \sin\theta = 0 \). In terms of unit circle, "cotangent" is derived from the coordinates of a point \( (\cos \theta, \sin \theta) \) on the circle.

It provides a useful way of expressing angles of a triangle in terms of ratios, especially when working with trigonometric identities. Understanding cotangent is crucial when solving trigonometric equations, as it often simplifies the computation or transformation of equations.

Cosecant

The cosecant function, abbreviated as "csc," is another important trigonometric function. It is the reciprocal of the sine function, and is expressed as:

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

Cosecant is undefined when \( \sin \theta = 0 \) because it involves division by zero. In the context of the unit circle, it is related to the y-coordinate and is specifically useful when dealing with right triangles or circular function relationships.

Recognizing how "cosecant" relates to sine helps in transforming and solving trigonometric equations effectively. Like cotangent, it also has periodic properties that are important when solving for full rotations or cycles of a trigonometric function.

Quadratic Equations

Quadratic equations form an essential part of algebra that also appear in trigonometry through the use of trigonometric identities. A standard quadratic equation can be represented as:

• \( ax^2 + bx + c = 0 \)

In trigonometry, these quadratic forms can incorporate functions like sine or cosine. Solving them might involve factoring, using the quadratic formula, or completing the square. The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our context, it helps find solutions for equations like \( \sin^2 \theta + \sin \theta - 1 = 0 \), which emerge from trigonometric transformations. These solutions can then be used to find the associated angles within specific intervals.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angle values that satisfy a given equation, often within a specified interval. These problems require understanding trigonometric identities to simplify the equation to a manageable form. Key strategies include:

• Using identities like \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Transforming the equation to a fundamental form
• Applying algebraic principles such as factoring

After simplification, the equation may become a simple algebraic or quadratic form, enabling the use of the quadratic formula if needed. Once a solution in terms of a trigonometric function (e.g., \( \sin \theta = x \)) is found, it can be used to calculate the specific angles using inverse functions.

Bear in mind trigonometric periodicity, as it allows for multiple solutions within the range (0° to 360° in this case). Each solution should be verified by plugging it back into the original equation to ensure it holds true.