Question

Find the exact value of the trigonometric functions at the indicated angle. \(\sin x, \cot x\), and \(\csc x\) for \(x=5 \pi / 6\)

Short answer

For \(x = \frac{5\pi}{6}\), the exact values of the trigonometric functions are: \[\sin(x) = \frac{1}{2}, \quad \cot(x) = -\sqrt{3}, \quad \text{and} \quad \csc(x) = 2.\]

Step-by-step solution

Identify the Position of the Angle on the Unit Circle

Let's first place the angle \(x = \frac{5\pi}{6}\) on the unit circle, so we can identify its reference angle and quadrant. The angle \(\frac{5\pi}{6}\) is equal to \(\frac{5(180^\circ)}{6} = 150^\circ\). Since the angle is between \(90^\circ\) and \(180^\circ\), it lies in the second quadrant.

Get the Reference Angle

Now let's find the reference angle by subtracting the angle between the second quadrant and the x-axis from the given angle: \( \alpha = 180^\circ - 150^\circ = 30^\circ \). The reference angle between the terminal side and the x-axis is \(\alpha = 30^\circ\). This reference angle has special properties, and we can use it to find the unknown trigonometric functions.

Find \(\sin(x)\) using the Reference Angle

In a 30-60-90 reference triangle, the sides have a ratio of 1 : 2 : \(\sqrt{3}\), where 1 is the length of the side opposite the 30-degree angle, 2 is the length of the hypotenuse, and \(\sqrt{3}\) is the length of the side opposite the 60-degree angle. From the definition of sine, we have: \( \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \). So, for a 30-degree angle, \( \sin(30^\circ) = \frac{1}{2} \). Since the angle \(x\) is in the second quadrant, the \(x\)-coordinate is negative while the \(y\)-coordinate is positive. Therefore, \(\sin(x) = \sin(\frac{5\pi}{6}) = \frac{1}{2}\).

Find \(\cot(x)\) using the Reference Angle

To find the \(\cot(x)\), let's first find \(\tan(x)\), since \(\cot(x) = \frac{1}{\tan(x)}\). In the 30-60-90 reference triangle, using the definition of tangent, we have: \( \tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}} \). So, for a 30-degree angle, \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Since the angle \(x\) is in the second quadrant, the tangent function is negative. Therefore, \(\tan(x) = -\tan(\frac{5\pi}{6}) = -\frac{1}{\sqrt{3}}\). Now, since \(\cot(x) = \frac{1}{\tan(x)}\), we can divide 1 by the tangent value: \( \cot(x) = -\sqrt{3} \).

Find \(\csc(x)\) using the Value of \(\sin(x)\)

Since the cosecant function is the reciprocal of the sine function, we can easily find it as: \( \csc(x) = \frac{1}{\sin(x)} \). Using the sine value we calculated in Step 3: \( \csc(x) = \csc(\frac{5\pi}{6}) = \frac{1}{\frac{1}{2}} = 2 \). Finally, we have found the exact trigonometric values: \(\sin(x) = \frac{1}{2}\), \(\cot(x) = -\sqrt{3}\), and \(\csc(x) = 2\).