Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution in \([0, 2\pi)\) is \( \theta = \frac{\pi}{12} \).

Step 1: Identify the Trigonometric Formulas

Observe that the equation \( \cos \theta - \sin \theta = \sqrt{2} \sin \frac{\theta}{2} \) involves \( \cos \theta \) and \( \sin \theta \) on one side, and \( \sin \frac{\theta}{2} \) on the other side. This suggests the use of double-angle formulas because the half-angle on one side can relate to full angles on the other.

Step 2: Transform \( \cos \theta - \sin \theta \)

Express \( \cos \theta - \sin \theta \) in a form that uses half-angles. One way is through a known identity: \( \cos \theta - \sin \theta = \sqrt{2}(\cos \theta \cos \frac{\pi}{4} - \sin \theta \sin \frac{\pi}{4}) = \sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right) \).

Step 3: Compare Both Sides

Now our equation looks like \( \sqrt{2} \cos\left(\theta + \frac{\pi}{4}\right) = \sqrt{2} \sin \frac{\theta}{2} \). By simplifying, eliminate \( \sqrt{2} \) from both sides, we have \( \cos\left(\theta + \frac{\pi}{4}\right) = \sin \frac{\theta}{2} \).

Step 4: Use Trigonometric Identity

Using the identity \( \sin x = \cos(\frac{\pi}{2} - x) \), rewrite \( \sin \frac{\theta}{2} \) as \( \cos(\frac{\pi}{2} - \frac{\theta}{2}) \). This equates to \( \cos\left(\theta + \frac{\pi}{4} \right) = \cos(\frac{\pi}{2} - \frac{\theta}{2}) \).

Step 5: Solve for Angles

Using the identity \( \cos A = \cos B \), we find solutions for \( A = B + 2n\pi \) or \( A = -B + 2n\pi \) where \( n \) is an integer. Therefore, set \( \theta + \frac{\pi}{4} = \frac{\pi}{2} - \frac{\theta}{2} + 2n\pi \) or \( \theta + \frac{\pi}{4} = -\left(\frac{\pi}{2} - \frac{\theta}{2}\right) + 2n\pi \) and solve for \( \theta \).

Step 6: Calculate Specific Solutions

First equation becomes \( \theta + \frac{\pi}{4} = \frac{\pi}{2} - \frac{\theta}{2} \). Solving gives \( 3\theta = \frac{\pi}{4} \) so \( \theta = \frac{\pi}{12} \). The second equation \( \theta + \frac{\pi}{4} = -\frac{\pi}{2} + \frac{\theta}{2} \). Solving gives \( \frac{3\theta}{2} = -\frac{3\pi}{4} \) leading to \( \theta = -\frac{\pi}{2} \), but it is outside \([0, 2\pi)\). Rotate by adding \( 2\pi \) to get more solutions in the interval if required.

Step 7: Confirm Solution in Interval

Verify if the solution \( \theta = \frac{\pi}{12} \) is indeed between \( 0 \) and \( 2\pi \). If any other solutions within one full rotation appear, confirm they satisfy the equation.

Key Concepts

Double-angle formulas

Double-angle formulas are essential tools in trigonometry. They allow us to express trigonometric functions of twice an angle in terms of functions of a single angle. These formulas are particularly beneficial when solving equations involving sine, cosine, and tangent with angles like \(2\theta\). There are three main double-angle formulas:

• \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
• \(\cos(2\theta) = 2\cos^2(\theta) - 1\)
• \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)

These equations provide the flexibility to work in different forms depending on the given problem. Similarly, the double-angle formulas for sine and tangent are given by:

• \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
• \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\)

In the context of solving equations, these identities are useful when transforming trigonometric expressions to achieve a simpler or more comparable form.

Half-angle formulas

Half-angle formulas are used when you need to express trigonometric functions of half-angles, or \(\frac{\theta}{2}\), in terms of the full angle \(\theta\). These formulas are particularly handy in deriving angles or simplifying complex trigonometric identities.

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

The signs \(+\) or \(-\) depend on the quadrant in which \(\frac{\theta}{2}\) lies. For instance, with the equation \(\cos\theta - \sin\theta = \sqrt{2}\sin\frac{\theta}{2}\), these formulas allow a transformation into terms involving full angles, aiding in finding solutions within certain intervals.

Unit circle

The unit circle is a fundamental concept in trigonometry that provides a geometric representation of sine and cosine functions. A unit circle is a circle with a radius of one centered at the origin of a coordinate plane. As you move around the circle, each point corresponds to an angle \(\theta\), measured in radians, from the positive x-axis.

• The x-coordinate of any point on the unit circle is \(\cos(\theta)\).
• The y-coordinate is \(\sin(\theta)\).
• The circle shows that \(\cos^2(\theta) + \sin^2(\theta) = 1\), a primary trigonometric identity.
• This concept is especially useful for understanding periodic behavior and cyclic properties of trigonometric functions.

Using the unit circle, we can visualize how trigonometric identities relate to the position of coordinates on this circle's circumference. It helps in determining angles and solving equations through graphical means, which can sometimes highlight symmetry or patterns not immediately obvious algebraically.

Solving trigonometric equations

Solving trigonometric equations involves finding the angles for which the equation holds true. This usually requires an understanding of trigonometric identities and formulas, like the ones we've seen with half-angle and double-angle equations. Here are general steps you should follow:

• Identify which trigonometric identities or formulas may simplify the equation.
• Transform the equation, if needed, into a form where known formulas can be applied.
• Use algebra to isolate the trigonometric function or combine identities to express one function in terms of another.
• Once simplified, determine the angles (or solutions) that satisfy the simplified equation.
• Ensure that solutions are within the given interval, such as \([0, 2\pi)\).
• Verify the solutions by plugging them back into the original equation, ensuring they satisfy all conditions.

For instance, by utilizing double-angle and half-angle formulas, the given exercise simplifies, leading to possible solutions like \(\theta = \frac{\pi}{12}\). Clear understanding of solving techniques ensures accuracy and effectively manages the range of the solution.