Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. cos θ = √3 2

검증된 단계별 안내

Recognize that the equation is \( \cos \theta = \frac{\sqrt{3}}{2} \). This is a standard cosine value corresponding to special angles on the unit circle.

Recall the unit circle values where \( \cos \theta = \frac{\sqrt{3}}{2} \). These occur at \( \theta = 30^\circ \) and \( \theta = 330^\circ \) within the interval \( [0^\circ, 360^\circ) \).

Understand that cosine is positive in the first and fourth quadrants, which is why these two angles are solutions.

Write the general solution for \( \cos \theta = \frac{\sqrt{3}}{2} \) as \( \theta = 30^\circ + 360^\circ k \) and \( \theta = 330^\circ + 360^\circ k \), where \( k \) is any integer.

Since the problem restricts \( \theta \) to the interval \( [0^\circ, 360^\circ) \), select the values \( \theta = 30^\circ \) and \( \theta = 330^\circ \) as the final solutions.

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주요 개념

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Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in degrees or radians, and their corresponding points on the unit circle determine the values of sine and cosine. Understanding how angles correspond to points on the unit circle helps identify where cosine values occur within a given interval.

Cosine Function and Its Values

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. Knowing the specific cosine values, such as cos θ = √3/2, helps identify standard angles (like 30° and 330°) where this value occurs. Recognizing these common values is essential for solving trigonometric equations.

Solving Trigonometric Equations in a Given Interval

When solving equations like cos θ = √3/2 over [0°, 360°), it is important to find all angles within the interval that satisfy the equation. Since cosine is positive in the first and fourth quadrants, solutions must be identified accordingly. This involves understanding the symmetry and periodicity of the cosine function.