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

Verified step by step guidance

1. insert step 1> Start by recognizing that the equation \( \cos \theta = \frac{\sqrt{3}}{2} \) is a standard trigonometric value.
2. insert step 2> Recall that \( \cos \theta = \frac{\sqrt{3}}{2} \) corresponds to angles where the cosine value is positive.
3. insert step 3> Identify the reference angle for \( \cos \theta = \frac{\sqrt{3}}{2} \), which is \( 30^\circ \) or \( \frac{\pi}{6} \) radians.
4. insert step 4> Determine the quadrants where cosine is positive: Quadrant I and Quadrant IV.
5. insert step 5> Find the angles in the specified interval [0°, 360°) that correspond to the reference angle in these quadrants: \( \theta = 30^\circ \) and \( \theta = 330^\circ \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles, allowing for easy identification of angle measures based on function values.

Cosine Function

The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In the context of the unit circle, the cosine of an angle θ is the x-coordinate of the point on the circle corresponding to that angle. Understanding the properties of the cosine function, including its periodicity and symmetry, is essential for solving equations involving cosine values.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to determine the angle that corresponds to a given trigonometric value. For example, if cos θ = √3/2, the arccosine function can be applied to find the angle θ. It is important to consider the range of the inverse functions and the specific interval given in the problem to find all possible angle solutions.