Problem 25 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\cot 510^{\circ}\)

Step-by-Step Solution

Answer

\(\cot 510^\circ = -\sqrt{3}\).

Step 1: Convert Angle to Standard Position

Given the angle of \(510^\circ\), we first need to convert it to a standard position by finding its coterminal angle between \(0^\circ\) and \(360^\circ\). We do this by subtracting \(360^\circ\) from \(510^\circ\) until the result is within the standard range. \(510^\circ - 360^\circ = 150^\circ\) which is in the desired range.

Step 2: Understand the Cotangent Function

The cotangent function, \(\cot\theta\), is defined as \(\frac{1}{\tan\theta}\), where \(\tan\theta\) is the tangent of the angle. Thus, \(\cot\theta = \frac{\cos\theta}{\sin\theta}\). Now, we need to find the sine and cosine of \(150^\circ\).

Step 3: Evaluate Trigonometric Values at 150°

At \(150^\circ\), the angle is in the second quadrant where sine is positive and cosine is negative. The reference angle for \(150^\circ\) is \(30^\circ\). Thus, \(\sin 150^\circ = \frac{1}{2}\) and \(\cos 150^\circ = -\frac{\sqrt{3}}{2}\).

Step 4: Calculate Cotangent Value

Using the sine and cosine values, calculate \(\cot 150^\circ = \frac{\cos 150^\circ}{\sin 150^\circ} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}\).

Step 5: Conclusion

The cotangent of \(510^\circ\), using the coterminal angle of \(150^\circ\), is \(-\sqrt{3}\).

Key Concepts

Coterminal Angles

Coterminal angles are angles which share the same initial and terminal sides but may differ by full rotations. This means if you keep adding or subtracting full rotations of \(360^\circ\) to an angle, the resulting angles are coterminal with each other.

In the current exercise, you start with \(510^\circ\). To find a coterminal angle within the standard position, which ranges from \(0^\circ\) to \(360^\circ\), you subtract \(360^\circ\). In this case:

• \(510^\circ - 360^\circ = 150^\circ\)

This means \(510^\circ\) is coterminal with \(150^\circ\). Coterminal angles such as these are essential when evaluating trigonometric functions, as they allow us to reduce complex angles to simpler, familiar ones.

Cotangent Function

The cotangent function, denoted by \(\cot\theta\), is one of the six basic trigonometric functions. It is specifically defined as the reciprocal of the tangent function:

\[ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} \]

This means you need both the cosine and sine values of an angle to find its cotangent value. For \(150^\circ\) from our exercise, you will use these sine and cosine values:

• \(\sin 150^\circ\) and \(\cos 150^\circ\)

Understanding the cotangent function is crucial since it helps in solving various trigonometric expressions and equations. It provides insight into the ratio of the adjacent side to the opposite side in a right triangle when interpreting through its geometric context.

Sine and Cosine Values

Sine and cosine values for angles are fundamental in trigonometry. They determine the behavior of the waveforms we commonly encounter. For an angle such as \(150^\circ\), you use its reference angle to find its sine and cosine values.

The reference angle for \(150^\circ\) is \(30^\circ\), and since \(150^\circ\) is in the second quadrant, the following rules apply:

• Sine is positive: \(\sin 150^\circ = \frac{1}{2}\)
• Cosine is negative: \(\cos 150^\circ = -\frac{\sqrt{3}}{2}\)

Knowing these values allows you to compute other related trigonometric functions, such as the cotangent in our exercise. These are not just mathematical tools but also aid in solving real-world problems involving wave patterns, oscillations, and rotations.