Problem 107 - Precalculus with Limits

Question

In Exercises \(105-108,\) use the trigonometric substitution to write the algebraic equation as a trigonometric equation of \(\theta\) where \(-\pi / 2<\theta<\pi / 2 .\) Then find \(\sin \theta\) and \(\cos \theta\). $$ 2 \sqrt{2}=\sqrt{16-4 x^{2}}, \quad x=2 \cos \theta $$

Step-by-Step Solution

Answer

The solution is \( \sin \theta = \sqrt{2}/2 \) and \( \cos \theta = 1/ \sqrt{2} \).

Step 1: Substitute

We start by substituting \( x = 2 \cos \theta \) into the given equation. That gives us \( 2 \sqrt{2} = \sqrt{16 - 4(2 \cos \theta)^2} \).

Step 2: Simplify the equation

Next, work through the equation and simplify it by performing operations. You should get \( 2 \sqrt{2} = \sqrt{16 - 16\cos^2 \theta } \) and further simplification gives \( 2 \sqrt{2} = 4 | \sin \theta |\).

Step 3: Solve for sin theta

We can now solve for \( \sin \theta \) by dividing both sides by 4 and we get \( \sin \theta = \sqrt{2}/2 \). We pick the positive root, as we are in the range \( -\pi/2 \) < \( \theta \) < \( \pi/2 \).

Step 4: Solve for cos theta

We already know from step 1 that \( x = 2 \cos \theta \) and we can use \( \cos \theta = x / 2 \) to get \( \cos \theta = \pm 1/ \sqrt{2} \). But since we are in the range \( -\pi/2 \) < \( \theta \) < \( \pi/2 \), the cosine function is positive, so \( \cos \theta = 1/ \sqrt{2} \).

Key Concepts

Solving Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions that we aim to solve for angles, typically given in radians or degrees. When students tackle these equations, they often face challenges in understanding the constraints of different trigonometric functions. For example, knowing that the sine function ranges from -1 to 1 can help students eliminate impossible values.

To solve a trigonometric equation, one must first isolate the trigonometric function. This often involves using algebraic manipulations such as factoring, distributing, or combining like terms. Once the trigonometric function is isolated, you can solve for the angle by applying inverse trigonometric functions. However, keep in mind the domain restrictions; such as in the exercise provided, the angle is restricted between \( -\pi/2 \) and \( \pi/2 \), which affects the potential solutions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values within the domain of the trigonometric functions. These identities are invaluable tools for simplifying and solving trigonometric expressions and equations. Familiarity with these fundamental identities, especially the Pythagorean identities, such as \( \sin^2\theta + \cos^2\theta = 1 \) and \( 1 + \tan^2\theta = \sec^2\theta \), is crucial for success in trigonometry.

In the given exercise, simplifying the equation yields \( 4\sin^2\theta \), which is derived from the Pythagorean identity by substituting \( \cos^2\theta \) with \( 1 - \sin^2\theta \). Recognizing this substitution allows you to transform the equation into a form where \( \sin\theta \) is isolated for solving.

Conversions Between Algebraic and Trigonometric Expressions

The connection between algebra and trigonometry is emphasized through conversions between algebraic and trigonometric expressions. These skills are crucial for manipulating and solving complex equations. In our exercise, the substitution \( x = 2\cos\theta \) creates a bridge between an algebraic expression \( \sqrt{16 - 4x^2} \) and a trigonometric expression \( 4|\sin\theta| \) by utilizing the Pythagorean theorem.

When converting expressions, it's important to visualize the unit circle and understand the relationships between the sides of a right triangle and trigonometric functions. This comprehension aids in substituting trigonometric functions into algebraic equations seamlessly, thereby creating equations that are often more straightforward to solve.