Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
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cos x = √3/2

검증된 단계별 안내

Identify the trigonometric equation given: \(\cos x = \frac{\sqrt{3}}{2}\). We need to find all values of \(x\) in the interval \([0, 2\pi)\) that satisfy this equation.

Recall that on the unit circle, \(\cos x\) corresponds to the x-coordinate of the point where the terminal side of angle \(x\) intersects the circle.

Determine the reference angle whose cosine is \(\frac{\sqrt{3}}{2}\). This is a common value corresponding to an angle of \(\frac{\pi}{6}\) radians (or 30°).

Since cosine is positive in the first and fourth quadrants, find the two angles in \([0, 2\pi)\) where \(\cos x = \frac{\sqrt{3}}{2}\). These are \(x = \frac{\pi}{6}\) and \(x = 2\pi - \frac{\pi}{6}\).

Write the solution set as \(\left\{ \frac{\pi}{6}, \frac{11\pi}{6} \right\}\), which includes all angles in the given interval where the cosine equals \(\frac{\sqrt{3}}{2}\).

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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. It helps relate angles to coordinates, where the x-coordinate corresponds to cosine and the y-coordinate to sine of the angle. Angles are measured in radians (0 to 2π) or degrees (0° to 360°), providing a framework to find trigonometric values.

Cosine Function and Its Values on the Unit Circle

Cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. For cos x = √3/2, we identify all angles where the x-coordinate equals √3/2. These angles typically occur in the first and fourth quadrants, reflecting the positive cosine values.

Solving Trigonometric Equations Over a Given Interval

Solving cos x = √3/2 over [0, 2π) means finding all angles within one full rotation where the cosine equals √3/2. Since cosine is positive in the first and fourth quadrants, solutions include two angles. Understanding the periodicity and symmetry of cosine is essential to list all valid solutions.

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°). <IMAGE> cos x = √3/2