Problem 46 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cos \theta \sqrt{1+\tan ^{2} \theta}=1 $$

Step-by-Step Solution

Answer

All \(\theta\) except \(90^\circ + k\cdot 180^\circ\), where \(k\) is an integer.

Step 1: Recognize the Identity

The equation \(\cos \theta \sqrt{1+\tan^{2} \theta}=1\) involves the trigonometric identity \(1+\tan^{2}\theta = \sec^{2}\theta\). This identity simplifies by recognizing that \(\tan^2 \theta + 1\) is equal to \(\sec^2 \theta\).

Step 2: Simplify Using Identity

Substitute \(\sec^2 \theta\) for \(1 + \tan^2 \theta\) in the radical, giving us \(\cos \theta \sqrt{\sec^2 \theta} = 1\). Since \(\sqrt{\sec^2 \theta} = \sec \theta\), it becomes \(\cos \theta \cdot \sec \theta = 1\).

Step 3: Solve for Angles

The equation \(\cos \theta \cdot \sec \theta = 1\) simplifies to \(\cos \theta \cdot \frac{1}{\cos \theta} = 1\), which is \(1 = 1\). This implies that the identity is satisfied for all angles where \(\cos \theta\) does not make the expression undefined.

Step 4: Ensure Domain Validation

Since this equation is equivalent to an identity, \(\theta\) can be any angle where \(\cos \theta eq 0\). This excludes \(\theta = 90^\circ + k\cdot 180^\circ\), where \(k\) is any integer, because \(\cos \theta\) would be zero, making \(\sec \theta\) undefined.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying and solving trigonometric equations. These identities are derived from the relationships between different trigonometric functions. In the given exercise, we encounter the identity \(1 + \tan^2 \theta = \sec^2 \theta\). This identity is a rearrangement of the Pythagorean identity applied in trigonometry.
Trigonometric identities help simplify complex trigonometric expressions by substituting equivalent expressions, thereby revealing hidden relationships. When you recognize an identity within an equation, like the one above, it can serve as a critical step towards simplifying and solving it.
Here are a few core trigonometric identities worth remembering:

• Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
• Reciprocal Identities: \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\)
• Tangent Identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Using these identities effectively can transform a difficult equation into something much more manageable.

Solutions of Trigonometric Equations

Solving trigonometric equations involves finding all possible values of the variable that make the equation true. For our specific problem, after utilizing the identity \(1 + \tan^2 \theta = \sec^2 \theta\), the equation simplifies to \(\cos \theta \cdot \frac{1}{\cos \theta} = 1\), which obviously holds true whenever \(\cos \theta eq 0\).
When determining the solutions for trigonometric equations, follow these general steps:

• Recognize and apply appropriate trigonometric identities to simplify the equation.
• Isolate the trigonometric function if possible or equate it to a known function value.
• Consider all initial angles and their respective general solutions.

The simplicity of this specific equation means that any angle where \(\cos \theta\) is non-zero will satisfy it. Thus, the solution set is extensive, except for the angles where \(\cos \theta\) equals zero, as outlined in the domain restrictions.

Domain Restrictions

Understanding domain restrictions is crucial when solving trigonometric equations as they define where the trigonometric function is valid. In trigonometric equations like \(\cos \theta \sqrt{1+\tan ^{2} \theta}=1\), you must ensure the values satisfy the domains of involved functions.
For example, \(\sec \theta\) is undefined where \(\cos \theta = 0\) because this would lead to division by zero, which is mathematically invalid. Therefore, the values of \(\theta\) where \(\cos \theta\) becomes zero are critical domain restrictions. In this case, those angles are \(90^\circ + k \cdot 180^\circ\), where \(k\) is any integer.
This exercise shows that imposing domain restrictions narrows down the valid solutions and maintains the integrity of the solution set. When solving equations, always check your work to confirm it respects these necessary constraints.