Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \cos 2 \theta+2 \cos ^{2} \theta=2 $$

Step-by-Step Solution

Answer

\(\theta = 30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}\).

Step 1: Transform the Equation

Start by simplifying and transforming the given equation \(\cos 2\theta + 2\cos^2\theta = 2\). Use the double angle identity \(\cos 2\theta = 2\cos^2\theta - 1\) to substitute into the equation. This results in \(2\cos^2\theta - 1 + 2\cos^2\theta = 2\).

Step 2: Combine and Simplify Terms

Combine the \(\cos^2\theta\) terms. This gives \(4\cos^2\theta - 1 = 2\). Next, add 1 to both sides to obtain \(4\cos^2\theta = 3\).

Step 3: Solve for \(\cos^2\theta\)

Divide both sides by 4 to isolate \(\cos^2\theta\). This results in \(\cos^2\theta = \frac{3}{4}\).

Step 4: Find \(\cos \theta\)

Take the square root of both sides. Recall that cosine can be both positive and negative in certain quadrants. So, \(\cos \theta = \pm \frac{\sqrt{3}}{2}\).

Step 5: Determine All Angles \(\theta\)

Find all possible \(\theta\) within the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\). For \(\cos \theta = \frac{\sqrt{3}}{2}\), \(\theta = 30^{\circ}\) and \(\theta = 330^{\circ}\). For \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\theta = 150^{\circ}\) and \(\theta = 210^{\circ}\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. In trigonometry, cosine, often abbreviated as cos, is used to measure how much projection a line makes on the horizontal axis, given an angle. It's a way to relate the angle with the length of the adjacent side in right-angled triangles. Some key features of the cosine function include:

• It takes any value in degrees or radians and outputs a value between -1 and 1.
• Cosine is known for its periodic nature, with a period of 360° (or 2π radians) meaning it repeats its values every 360°.
• Cosine of 0° is 1, while cosine of 90° is 0, and it continues similarly around the unit circle.

Understanding the cosine function is crucial when delving into solving trigonometric equations, such as the one given in the exercise. Here, identifying the values of cosine at specific angles helps to determine solutions for the equation.

Double Angle Identity

The double angle identities are a set of formulas that express trigonometric functions of double angles, in terms of single angles. These identities are extremely useful in simplifying trigonometric expressions and solving equations. For cosine, the double angle identity is as follows: \[ \cos 2\theta = 2\cos^2\theta - 1 \] This formula is pivotal because it allows transformation of an equation involving \(\cos 2\theta\) into one that involves just \(\cos\theta\).

• The formula reveals how the cosine of double an angle relates to the square of the cosine of the angle itself.
• It is particularly useful in converting equations to a simpler form where values of \(\theta\) can be directly determined.

In the problem provided, the double angle identity helped to reduce the complexity of the equation, making it more manageable to combine and solve.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy a given trigonometric equation. These solutions often lie within a specified range, such as from 0° to 360°.To successfully solve these equations, follow these steps:

• Simplify the equation using known trigonometric identities, like double angle or Pythagorean identities. This helps to express the equation in terms of a single trigonometric function.
• Rearrange and solve for the trigonometric function. This might involve extracting square roots or isolating the function on one side.
• Determine all angle solutions within the given interval by considering the unit circle or specific trigonometric values.

For the given problem, converting the equation using the double angle identity and simplifying led to finding \(\cos \theta = \pm \frac{\sqrt{3}}{2}\). This provides two sets of solutions for \(\theta\), each corresponding to potential angles where the cosine function achieves these values within the given range.