Problem 69 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\cos \theta=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

The possible values of \( \theta \) are \( 30^{\circ} \) and \( 330^{\circ} \).

Step 1: Understand the Problem

We need to find all possible values of \( \theta \) where the cosine of \( \theta \) equals \( \frac{\sqrt{3}}{2} \). These angles must be in the range of \( 0^{\circ} \) to \( 360^{\circ} \).

Step 2: Recall Important Values

The cosine of certain standard angles is well-known. Particularly, \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). Hence, \( 30^{\circ} \) is a candidate value for \( \theta \).

Step 3: Determine all Angles in the Range

The cosine function is positive in the first and fourth quadrants. Thus, the possible values of \( \theta \) are at \( 30^{\circ} \) (first quadrant) and \( 360^{\circ} - 30^{\circ} = 330^{\circ} \) (fourth quadrant).

Step 4: Verify the Solution

Verify that both possible values satisfy the initial equation. \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) and \( \cos(330^{\circ}) = \frac{\sqrt{3}}{2} \). Thus, \( \theta = 30^{\circ} \) and \( \theta = 330^{\circ} \) are solutions.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometric equations. They help you understand relationships between sine, cosine, tangent, and their reciprocal functions. Knowing these identities allows students to manipulate and simplify expressions. For this problem, the focus is on the cosine function. One of the most notable identities is the Pythagorean identity, which states:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\) in a right triangle

Remembering these and other trigonometric identities helps you quickly recall the cosine of common angles, like 30°, 45°, 60°, among others. Recognizing that these identities are deeply interconnected broadens your problem-solving skills and enhances your trigonometry knowledge.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. Cosine has a wave-like pattern, oscillating between -1 and 1 over a cycle of 360° (or 2π radians).
### Graph and Properties- **Amplitude:** The maximum value of the cosine function is 1.- **Period:** The function completes one full cycle every 360°. - **Even Function:** Cosine is symmetrical about the y-axis, meaning \(\cos(-\theta) = \cos(\theta)\).
In the context of our equation \(\cos \theta = \frac{\sqrt{3}}{2}\), identifying where the cosine of an angle corresponds to this specific value means we look for specific angles on the cosine wave. Recognizing that cosine is positive in the first and fourth quadrants helps us pinpoint possibly relevant angles, such as 30° and 330°.

Angle Measurement

Angles can be measured in degrees or radians, which are the two most common units. To effectively solve trigonometric equations, being comfortable with converting between these units is crucial, as they are often interchangeable in mathematics.

• Degrees: A complete circle is 360°.
• Radians: A complete circle is \(2\pi\) radians. Therefore, \(180° = \pi\) radians.

Knowing these conversions allows for flexibility, especially when angles are used in deeper mathematical analysis, calculus, or real-world applications. For the exercise's context, recognizing that specific angles like 30° (\(\frac{\pi}{6}\) radians) fall within known ranges on the cosine function, grants clarity on which angles to choose. Understanding how angle measurement affects functions like cosine ensures precise solutions and improved comprehension in broader trigonometry contexts.