The unit circle has been divided into twelve equal arcs, corresponding to t-values of


0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋


Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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cos 5𝜋/6

Verified step by step guidance

1. Recall that on the unit circle, the coordinates of a point corresponding to an angle \(t\) are given by \((\cos t, \sin t)\).
2. Identify the angle \(t = \frac{5\pi}{6}\) on the unit circle. This angle is located in the second quadrant, where cosine values are negative and sine values are positive.
3. Use the known exact coordinates for \(\frac{5\pi}{6}\). Since the circle is divided into twelve equal arcs of \(\frac{\pi}{6}\) each, the coordinates at \(\frac{5\pi}{6}\) correspond to the point \(\left(-\cos \frac{\pi}{6}, \sin \frac{\pi}{6}\right)\).
4. Recall the exact values: \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin \frac{\pi}{6} = \frac{1}{2}\). Substitute these into the coordinates to get \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).
5. Since cosine corresponds to the x-coordinate on the unit circle, the value of \(\cos \frac{5\pi}{6}\) is the x-coordinate of this point, which is \(-\frac{\sqrt{3}}{2}\).

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are often measured in radians, where 2π radians correspond to a full rotation of 360°. Dividing the circle into twelve equal arcs means each arc measures π/6 radians, providing standard angle measures for evaluating trigonometric functions.

Coordinates on the Unit Circle and Trigonometric Functions

Each point on the unit circle corresponds to an angle t and has coordinates (x, y), where x = cos(t) and y = sin(t). These coordinates allow direct evaluation of sine and cosine values for given angles. For example, cos(5π/6) is the x-coordinate of the point at angle 5π/6 on the unit circle.

Domain and Undefined Values of Trigonometric Functions

While sine and cosine are defined for all real numbers, some trigonometric functions like tangent, cotangent, secant, and cosecant can be undefined at certain angles where their denominators are zero. Understanding when a function is undefined is crucial for correctly interpreting values from the unit circle.