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Chapter 2: Problem 57

Find exact values for each of the following, if possible. \(\cot 45^{\circ}\)

Short Answer

Expert verified
The exact value of \( \cot 45^{\circ} \) is 1.

Step by step solution

01

Understanding cotangent

The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side. In terms of the sine and cosine functions, it is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
02

Identifying reference angle values

Since \( 45^{\circ} \) is a standard angle, we can use its common trigonometric values. At \( 45^{\circ} \), \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) and \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).
03

Calculating the cotangent

Use the formula for cotangent: \( \cot 45^{\circ} = \frac{\cos 45^{\circ}}{\sin 45^{\circ}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \).

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Key Concepts

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

Cotangent
Cotangent is one of the less commonly encountered trigonometric functions, but it's essential for certain applications.
Cotangent, often abbreviated as \( \cot \), is the reciprocal of the tangent function in trigonometry.
This means it is the ratio of the adjacent side to the opposite side in a right triangle.
The formula for cotangent in terms of the basic trig functions is:
  • \( \cot \theta = \frac{1}{\tan \theta} \)
  • Alternatively, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
Understanding cotangent is particularly valuable in various geometry and calculus problems, helping simplify and solve trigonometric equations.
Trigonometric Ratios
Trigonometric ratios are fundamental in understanding relationships in right triangles.
They provide a way to relate angles to side lengths.
Let's explore the three primary ratios:
  • Sine (\(\sin\)): This ratio compares the length of the opposite side to the hypotenuse: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
  • Cosine (\(\cos\)): This is the ratio of the adjacent side to the hypotenuse: \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
  • Tangent (\(\tan\)): Tangent is the ratio of the opposite side to the adjacent side: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).
    • Cotangent, as discussed earlier, is the reciprocal of this: \(\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}\).
These ratios allow us to calculate any part of a right triangle as long as we know one side length and one angle.
Right Triangle
A right triangle is a type of triangle that includes a 90-degree angle.
This is the simplest form of a triangle and is foundational in trigonometry.
The right triangle helps us understand how trigonometric ratios are derived. The 90-degree angle, or right angle, is crucial because it creates a basis for defining the functions:
  • The side opposite the right angle is known as the hypotenuse. It is the longest side of the triangle.
  • The other two sides are called the opposite and adjacent sides, relative to a chosen angle in the triangle.
Using these relationships, trigonometric functions become practical tools for solving problems. Knowing any two pieces of information (two side lengths or one side length and an angle) allows us to determine the missing parts of the triangle. This application makes the right triangle one of the cornerstones in both pure and applied mathematics.

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Most popular questions from this chapter

Two people decide to find the height of an obelisk. They position themselves 25 feet apart in line with, and on the same side of, the obelisk. If they find that the angles of elevation from the ground where they are standing to the top of the obelisk are \(65^{\circ}\) and \(54^{\circ}\), how tall is the obelisk?

From here on, each Problem Set will end with a series of review problems. In mathematics, it is very important to review. The more you review, the better you will understand the topics we cover and the longer you will remember them. Also, there will be times when material that seemed confusing earlier will be less confusing the second time around. The problems that follow review material we covered in Section 1.2. Find \(x\) so that the distance between \((x, 2)\) and \((1,5)\) is \(\sqrt{13}\).

Triangle \(A B C\) is a right triangle with \(C=90^{\circ}\). If \(a=16\) and \(c=20\), what is \(\sin A\) ? a. \(\frac{3}{5}\) b. \(\frac{5}{3}\) c. \(\frac{4}{5}\) d. \(\frac{5}{4}\)

For each expression that follows, replace \(x\) with \(30^{\circ}, y\) with \(45^{\circ}\), and \(z\) with \(60^{\circ}\), and then simplify as much as possible. $$ 2 \sin x $$

A person standing on top of a 15 -foot high sand pile wishes to estimate the width of the pile. He visually locates two rocks on the ground below at the base of the sand pile. The rocks are on opposite sides of the sand pile, and he and the two rocks are in the same vertical plane. If the angles of depression from the top of the sand pile to each of the rocks are \(29^{\circ}\) and \(17^{\circ}\), how far apart are the rocks?
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