Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2-\sin 2 \theta=4 \sin 2 \theta$$

Step-by-Step Solution

Answer

\(\theta = 11.8^\circ + k \cdot 180^\circ\) or \(\theta = 78.2^\circ + k \cdot 180^\circ\).

Step 1: Rewrite the Equation in Terms of \(\sin 2\theta\)

The given equation is \(2 - \sin 2\theta = 4 \sin 2\theta\). First, rearrange the equation by moving all terms involving \(\sin 2\theta\) to one side of the equation:\[ 2 = 4 \sin 2\theta + \sin 2\theta \].Combine the \(\sin 2\theta\) terms:\[ 2 = 5 \sin 2\theta \].

Step 2: Solve for \(\sin 2\theta\)

Isolate \(\sin 2\theta\) by dividing both sides of \(2 = 5\sin 2\theta\) by 5:\[ \sin 2\theta = \frac{2}{5} \].

Step 3: Find General Solution for \(2\theta\)

Since \(\sin 2\theta = \frac{2}{5}\), use the inverse sine function to find \(2\theta\):\[ 2\theta = \arcsin\left( \frac{2}{5} \right) \].This gives the principal value:\[ 2\theta = \arcsin\left( \frac{2}{5} \right) \approx 0.4115 \text{ radians} \].The sine function is positive in both the first and second quadrants, so the general solution for \(2\theta\) is:\[ 2\theta = 0.4115 + 2k\pi \quad \text{or} \quad 2\theta = \pi - 0.4115 + 2k\pi \quad \text{for any integer } k \].

Step 4: Solve for \(\theta\)

Divide each equation in Step 3 by 2 to solve for \(\theta\):1. \(\theta = \frac{0.4115}{2} + k\pi = 0.2058 + k\pi\).2. \(\theta = \frac{\pi - 0.4115}{2} + k\pi = 1.3658 + k\pi\).These represent the angles \(\theta\) must satisfy.

Step 5: Provide Exact Solutions in Degrees

Convert the radian solutions for \(\theta\) to degrees for the specified range:- \(0.2058\) radians to degrees is approximately \(11.8^\circ\).- \(1.3658\) radians to degrees is approximately \(78.2^\circ\).Therefore, the general solutions in degrees are:- \(\theta = 11.8^\circ + k \cdot 180^\circ\)- \(\theta = 78.2^\circ + k \cdot 180^\circ\).

Key Concepts

Radians and Degrees

In trigonometry, angles can be measured in degrees or radians. Both are essential to understand, as they are used in different contexts. A full circle is 360 degrees or \(2\pi\) radians. This is crucial: degrees are based on dividing a circle into 360 parts, while radians measure angles based on the radius of a circle. To convert from degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Conversely, convert radians to degrees by \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

• Understanding radians is vital for solving problems involving trigonometric functions, especially when using calculus.
• Degrees are often used in geometry and for practical problems.

Familiarity with both can enhance problem-solving skills across different mathematical areas. Remember, learning to switch between these systems is like learning to be bilingual in math!

Inverse Trigonometric Functions

Inverse trigonometric functions help us to find angles when we know the ratios of the sides of right triangles. For example, the inverse sine function, known as \( \arcsin\), tells us the angle whose sine is a given number. This is crucial if you want to find specific angle values when given certain side ratios.

• \( \arcsin(x) \) will return an angle in the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
• Be aware: inverse functions only return principal values, the smallest angle solution.

Knowing the range of these functions is essential for interpreting their results correctly. In many cases, particularly in trigonometry problems, you'll have to use the principal value, then think about the general solution to find all possible angle values.

General Solution in Trigonometry

A general solution in trigonometry is an expression that gives all possible solutions to a trigonometric equation, not just the principal or smallest ones. When solving equations like \(2\theta = \arcsin\left( \frac{2}{5} \right)\), we need to consider periodic properties of trigonometric functions.

• Because sine is periodic with period \(2\pi\), solutions repeat every \(2\pi\).
• The general solution often involves adding \(2k\pi\) (or \(k \cdot 360^\circ\) in degrees) to cover every cycle \(k\).

That's why we calculate for both the first and second quadrants where sine is positive, and give formulas like \(2\theta = 0.4115 + 2k\pi\) and \(2\theta = \pi - 0.4115 + 2k\pi\). These expressions let you find all angles satisfying the original equation. This comprehensive approach is valuable because it completes the picture beyond numerical solutions.