Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$2 \sin \theta=1-2 \cos \theta$$

Step-by-Step Solution

Answer

The solutions are \(\theta = 114.3^{\circ}\) and \(\theta = 335.7^{\circ}\).

Step 1: Simplify and Rearrange the Equation

Start by rewriting the given equation: \[ 2 \sin \theta = 1 - 2 \cos \theta \]Rearrange this equation to bring all terms involving trigonometric functions to one side:\[ 2 \sin \theta + 2 \cos \theta = 1 \]

Step 2: Factor Out Common Terms

Factor out the 2 from the left side of the equation:\[ 2(\sin \theta + \cos \theta) = 1 \]Then divide both sides by 2:\[ \sin \theta + \cos \theta = \frac{1}{2} \]

Step 3: Use Trigonometric Identity

We can rewrite \(\sin \theta + \cos \theta\) using the identity:\[ \sin \theta + \cos \theta = \sqrt{2} \sin(\theta + 45^{\circ}) \]

Step 4: Substitute and Solve for θ

Substitute the identity into the equation:\[ \sqrt{2} \sin(\theta + 45^{\circ}) = \frac{1}{2} \]Solve for \(\sin(\theta + 45^{\circ})\):\[ \sin(\theta + 45^{\circ}) = \frac{1}{2} \times \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{8}} \approx 0.3536 \]

Step 5: Determine Angles in Standard Position

The arcsine function allows us to find the angle:\[ \theta + 45^{\circ} \approx \arcsin(0.3536) \approx 20.7^{\circ} \]Since the sine function is positive in the first and second quadrants, the solutions for\(\theta + 45^{\circ}\) over the interval \([0^{\circ}, 360^{\circ})\) are \(20.7^{\circ}\) and \(180^{\circ} - 20.7^{\circ} \approx 159.3^{\circ}\).

Step 6: Solve for θ

Now calculate \(\theta\):\[ \theta = 20.7^{\circ} - 45^{\circ} \approx -24.3^{\circ} \]Since \(-24.3^{\circ}\) isn’t in the interval \([0^{\circ}, 360^{\circ})\), add 360 to it:\[ \theta = 360^{\circ} - 24.3^{\circ} = 335.7^{\circ} \]Now for the second angle:\[ \theta = 159.3^{\circ} - 45^{\circ} = 114.3^{\circ} \]

Step 7: Verify Solutions

Verify that both \(114.3^{\circ}\) and \(335.7^{\circ}\) meet the original equation by substituting back. This ensures both angles satisfy the equation within the given interval.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometric equations as they allow us to rewrite equations in simpler or more convenient forms. In this exercise, the identity \( \sin \theta + \cos \theta = \sqrt{2} \sin(\theta + 45^{\circ}) \) was crucial in simplifying the problem. This specific identity comes from the combination of the sine and cosine functions being transformed into a single sine function, adjusted by a phase shift of 45 degrees.
This transformation can help break down complex expressions into simpler components, often revealing the underlying structure of an equation. Knowing how to apply these identities appropriately takes practice but can dramatically simplify solving trigonometric equations.

• Using identities helps in expressing trigonometric equations in alternate forms, facilitating easier manipulation and solutions.
• Memorizing and understanding common identities, such as double angle or sum and difference formulas, can be incredibly beneficial.

Interval Solutions

An important part of solving trigonometric equations is ensuring the solutions are within the specified interval. In this exercise, the required interval was \([0^{\circ}, 360^{\circ})\). Understanding the nature of the interval is crucial since trigonometric functions can repeat their values in cycles.
For instance, the equation in this exercise involved finding values of \(\theta\) where sine and cosine functions yield particular results. Consequently, adjustments were necessary when initial calculations gave negative angles, as all solutions needed to fit within the specified interval.

• When solving trigonometric equations, be sure to adjust angles that fall outside the desired interval by adding or subtracting 360 degrees, ensuring they fit within the specified range.
• Being aware of the periodic nature of trigonometric functions helps when identifying multiple solutions within a given interval.

Arcsine Function

The arcsine function, denoted as \( \arcsin \), is the inverse of the sine function. It is used to determine the angle whose sine is a given number, an essential step in solving trigonometric equations.
In this problem, \( \arcsin(0.3536) \approx 20.7^{\circ} \) was calculated to find the possible angles for the expression. Since the sine function is positive in the first and second quadrants, it was vital to consider both possible angles for \( \theta + 45^{\circ} \) in these quadrants.

• The range of the arcsine function is typically \([-90^{\circ}, 90^{\circ}]\), but when solving problems, consider all possible angles resulting in the given sine value within the full circle (first and second quadrants here).
• Understanding the periodic properties of trigonometric functions helps identify all potential solutions resulting from arcsin values within a specified interval.