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Chapter 6: Problem 5

In Exercises 1-12, find the exact value of each expression. Give the answer in radians. $$ \cot ^{-1}(-1) $$

Short Answer

Expert verified
The exact value is \\(\frac{3\pi}{4}\\) radians.

Step by step solution

01

Understanding the Inverse Cotangent Function

The function \( ic^{-1}(x)\) gives the angle \( heta\) such that \( heta\) has the cotangent value equal to \( x\). Thus, finding \( ext{cot}^{-1}(-1)\) means finding an angle in radians whose cotangent is equal to \( -1\).
02

Identify the Range for \\( ext{cot}^{-1}\\)

The range of the \( ext{cot}^{-1}\) function is from \( 0\) to \( ext{蟺}\), excluding \( ext{蟺}\). We need to find an angle in this range where the cotangent is \( -1\).
03

Finding Angles with Specific Cotangent Values

The cotangent of an angle is the reciprocal of its tangent. So if \( ext{cot}( heta) = -1\), then \( ext{tan}( heta) = -1\) as well. This occurs at specific angles, typically \( ext{3蟺/4}\) in this range, because \( an( ext{3蟺/4}) = -1\).
04

Verify the Angle

Since \( ext{3蟺/4}\) is within the range of \( 0\) to \( ext{蟺}\) and satisfies \( ext{cot}( ext{3蟺/4}) = -1\), it is the correct answer for \( ext{cot}^{-1}(-1)\).

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Key Concepts

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

Cotangent Function
The cotangent (\( \cot \)) function is a fundamental concept in trigonometry. It represents the reciprocal of the tangent function. In simpler terms, for an angle \( \theta \), the cotangent is defined as \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Breaking this down further:
  • The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the adjacent side, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
  • Therefore, the cotangent is the ratio of the adjacent side to the opposite side, which flips the tangent's ratio: \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \).
This relationship remains true in the unit circle where the cotangent can also be expressed using sine (\( \sin \)) and cosine (\( \cos \)): \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). Note that the cotangent function is undefined when \( \sin(\theta) = 0 \) because you cannot divide by zero.
Radians
Radians are a unit of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's circumference being equal to \( 2\pi \) radians. This means:
  • One full revolution around a circle is \( 2\pi \) radians, which is equivalent to 360 degrees.
  • This makes \( \pi \) radians equal to 180 degrees, and \( \frac{\pi}{2} \) radians equal to 90 degrees.
Using radians allows for a more natural and efficient formulation of mathematical and physical equations, especially those involving periodic functions like trigonometric ones. When working through problems like finding \( \cot^{-1}(-1) \), it is crucial to express angles in radians to match the conventional mathematical setting.
Angle Finding
Finding angles that satisfy trigonometric functions is often about understanding the function's behavior and its inverse relationship. In the context of \( \cot^{-1}(x) \):
  • We need to find an angle whose cotangent equals \( x \).
  • This angle must lie within the function's range, typically between \( 0 \) and \( \pi \) radians for \( \cot^{-1}(x) \).
The process requires familiarity with standard angles and their trigonometric values. For example, you might recognize that for \( \tan(\theta) = -1 \), the standard angle is \( \frac{3\pi}{4} \) since it's within the allowable range and understanding the relationship \( \cot(\theta) = -\tan(\theta) \), aids in identifying the correct angle.
Inverse Trigonometric Functions
Inverse trigonometric functions are essential in mathematics for determining angles with given trigonometric function values. The inverse cotangent function, \( \cot^{-1}(x) \), specifically finds the angle \( \theta \) whose cotangent is \( x \).
  • The notation \( \cot^{-1}(x) \) signifies the angle \( \theta \) where \( \cot(\theta) = x \).
  • The range of \( \cot^{-1}(x) \) is from \( 0 \) to \( \pi \) radians (excluding \( \pi \)).
A crucial part of using inverse trigonometric functions is remembering their restricted ranges. This restriction ensures that for each function value, there is a unique angle, making the function injective (one-to-one) within its range. These functions allow students to transition from trigonometric values back to angles, solving equations in geometry and calculus.

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Most popular questions from this chapter

Optics. Assume that light is going from air into a diamond. Calculate the refractive angle \(\theta_{r}\) if the incidence angle is \(\theta_{i}=75^{\circ}\), and the index of refraction values for air and diamond are \(n_{i}=1.00\) and \(n_{r}=2.417\), respectively. Round to the nearest degree. (See Example 8 for Snell's law.)

Solve for the smallest positive \(x\) that makes this statement true: $$ \sin \left(x+\frac{\pi}{4}\right)+\sin \left(x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2} $$

In Exercises 1-36, solve each of the trigonometric equations exactly on the interval \(0 \leq x<2 \pi\). $$ \csc x-\cot x=\frac{\sqrt{3}}{3} $$

In Exercises 37-48, solve each of the trigonometric equations on the interval \(0^{\circ} \leq \theta<360^{\circ}\). Give answers in degrees and round to two decimal places. $$ \frac{\sin ^{2} x-2 \sin x}{\tan x}=0 $$

In Exercises 1-36, solve each of the trigonometric equations exactly on the interval \(0 \leq x<2 \pi\). $$ \sin (2 x)=\sqrt{3} \sin x $$
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