Problem 52 - Algebra and Trigonometry

Question

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\sin \theta-\cos \theta=\frac{1}{2}\)

Step-by-Step Solution

Answer

Solve \( \sin \theta - \cos \theta = \frac{1}{2} \) with a trigonometric identity to find solutions in \([0, 2\pi)\).

Step 1: Utilize a Trigonometric Identity

Start by using the identity \( \sin \theta - \cos \theta = \sqrt{2} \sin(\theta - \pi/4) \). This is derived from expressing \( \sin \theta \) and \( \cos \theta \) as functions of \( \theta - \pi/4 \).

Step 2: Equate to Solve

Substitute the identity into the equation: \( \sqrt{2} \sin(\theta - \pi/4) = \frac{1}{2} \). Simplify to find \( \sin(\theta - \pi/4) = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \).

Step 3: Solve for the Angle

Now, solve \( \sin(\theta - \pi/4) = \frac{\sqrt{2}}{4} \) by finding the angles that satisfy the sine value. Use \( \arcsin\left(\frac{\sqrt{2}}{4}\right) \) to find the principal value of the angle.

Step 4: General Solutions

Recall that \( \sin x = k \) has solutions at \( x = \arcsin(k) + 2\pi n \) and \( x = \pi - \arcsin(k) + 2\pi n \) for integer \( n \). Compute these for our equation: \( \theta - \pi/4 = \arcsin\left(\frac{\sqrt{2}}{4}\right) + 2\pi n \) and \( \theta - \pi/4 = \pi - \arcsin\left(\frac{\sqrt{2}}{4}\right) + 2\pi n \).

Step 5: Solve for \( \theta \)

Add \( \pi/4 \) to both sides of each equation to solve for \( \theta \):- \( \theta = \arcsin\left(\frac{\sqrt{2}}{4}\right) + \pi/4 + 2\pi n \) - \( \theta = \pi - \arcsin\left(\frac{\sqrt{2}}{4}\right) + \pi/4 + 2\pi n \).

Step 6: Find Solutions in the Given Interval

Evaluate the above expressions for \( n = 0 \), \( n = 1 \), etc., to find which solutions fall within \([0, 2\pi)\). Calculate and check each possibility to ensure it lies in the interval.

Key Concepts

Double Angle Formula

The double angle formula in trigonometry helps us express trigonometric functions of double angles or simply relates an angle's trigonometric function with those of its double. It provides the foundation to solve complex trigonometric equations. The formulae are:

• For cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• Or alternatively: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \) or \( \cos(2\theta) = 1 - 2\sin^2(\theta) \)
• For sine: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• And for tangent: \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \)

These formulas are particularly useful in breaking down and solving equations where an angle is doubled, such as when dealing with transformations or simplifications in trigonometry. In solving equations, they provide a foundation to express given trigonometric expressions in a different form which often makes the problem easier to tackle.

Half Angle Formula

The half angle formulas are a group of trigonometric identities that express functions of half angles. They are derived from the double angle formulas, making them instrumental when solving trigonometric equations involving half angles of a given function. The formulas are:

• For sine: \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• For cosine: \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)
• For tangent: \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \)

These formulas help break down or simplify functions into more manageable parts. Whether you're working with integrals, derivatives, or solving complex trigonometric equations, the half-angle formulas can streamline your work. They also aid in finding exact values of trigonometric functions without using a calculator.

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that are true for every value of the occurring variable. They are equations like the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and are essential tools in simplifying and transforming trigonometric expressions. Some crucial ones include:

• Reciprocal identities: \( \csc(\theta) = \frac{1}{\sin(\theta)} \), \( \sec(\theta) = \frac{1}{\cos(\theta)} \), \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Quotient identities: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)
• Co-function identities for complementary angles: \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \)

For the given exercise, the identity used is \( \sin\theta - \cos\theta = \sqrt{2} \sin(\theta - \frac{\pi}{4}) \), which cleverly simplifies according to another angle transformation rule. Knowing these identities well can allow you to manipulate and solve equations more easily and recognize patterns.

Solving Equations

Solving trigonometric equations involves finding the values of the variables that satisfy the given equation. Here, we focus on reducing the problem to simpler forms, often via substitutions using the identities and formulas we've briefly discussed.

The steps for solving a problem like our original exercise include:

• Identify a suitable trigonometric identity or formula to simplify the equation.
• Substitute and rearrange terms to form an easily solvable expression.
• Compute the solution using inverse trigonometric functions, if necessary. For example, find \( \theta \) by applying \( \arcsin \) or \( \arccos \).
• Make use of general solutions of trigonometric functions to find all potential solutions, since trigonometric functions are periodic.
• Filter results to include only those solutions within the specified interval, such as \([0, 2\pi)\).

Because of the periodic nature, solutions often produce multiple possible angles for trigonometric equations. Practicing these methods will sharpen your problem-solving skills in trigonometry and help to approach problems methodically.