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Chapter 6: Problem 67

Find the reference angle and the exact function value if they exist. $$\cot \left(-90^{\circ}\right)$$

Short Answer

Expert verified
The cotangent of \(-90^{\text{掳}}\) is undefined.

Step by step solution

01

Recognize the Angle

Identify the given angle: \(-90^{\text{掳}}.\)
02

Reference Angle Calculation

To find the reference angle, convert the angle to a positive equivalent within a complete circle (0掳 to 360掳). Add 360掳 to \(-90^{\text{掳}}\): \(-90^{\text{掳}} + 360^{\text{掳}} = 270^{\text{掳}}.\) \Furthermore, the reference angle is the acute angle formed with the x-axis, which in this case is \(360^{\text{掳}} - 270^{\text{掳}} = 90^{\text{掳}}.\)
03

Evaluate the Cotangent Function

Find \(\text{cot}(270^{\text{掳}}). \) The cotangent function is the reciprocal of the tangent function, i.e., \(\text{cot}(x) = \frac{1}{\text{tan}(x)}. \) At \(270^{\text{掳}}, \text{tan}(270^{\text{掳}}) = \text{undefined}\), thus \(\text{cot}(270^{\text{掳}}) = \frac{1}{\text{undefined}} = 0.\)
04

Conclusion

Given \(\text{cot}(-90^{\text{掳}})\) is undefined due to the undefined state of tangent at \(270^{\text{掳}}.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
The cotangent function, often abbreviated as \(\text{cot}(x)\), is one of the lesser-known trigonometric functions. Like the sine, cosine, and tangent functions, it is derived from a right-angled triangle. Specifically, cotangent is defined as the reciprocal of the tangent function. Mathematically, this relationship is expressed as \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\). This means that if you know the value of \(\text{tan}(x)\), you can easily find the cotangent by taking its reciprocal.

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