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Chapter 11: Problem 30

Use a calculator and reciprocal relationships to find each ratio correct to four decimal places. $$\cot 67^{\circ}$$

Short Answer

Expert verified
The value of \( \cot 67^{\circ} \) is approximately 0.4245.

Step by step solution

01

Understand the Cotangent and Reciprocal Relationship

The cotangent of an angle in a right triangle is the reciprocal of the tangent of that angle. That is, \( \cot \theta = \frac{1}{\tan \theta} \). In this exercise, you need to find \( \cot 67^{\circ} \).
02

Use Calculator to Find Tangent of 67掳

Use a scientific calculator to find \( \tan 67^{\circ} \). Make sure your calculator is set to degree mode. After calculation, \( \tan 67^{\circ} \approx 2.3559 \).
03

Calculate the Cotangent

Since \( \cot 67^{\circ} = \frac{1}{\tan 67^{\circ}} \), compute the cotangent using the value obtained in Step 2: \( \cot 67^{\circ} = \frac{1}{2.3559} \approx 0.4245 \).
04

Verify the Result

Confirm the correctness of the result by using the reciprocal relation directly on a calculator: \( \frac{1}{2.3559} = 0.4245 \). The calculated value checks out.

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Key Concepts

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

Reciprocal Relationships
Reciprocal relationships in trigonometry help us connect different trigonometric functions. A reciprocal is simply the flipped version of a number or function. In simpler terms, for a given number, its reciprocal is one divided by that number.

For trigonometric functions like tangent and cotangent, this means:
  • The cotangent of an angle is the reciprocal of the tangent of that angle.
  • Expressed in a formula: if \( \tan \theta \) is a function, then \( \cot \theta = \frac{1}{\tan \theta} \).
Reciprocal relationships are particularly helpful when solving problems using trigonometric ratios. With a reciprocal, you can easily switch from one function to another, especially when using a calculator.
Cotangent
Cotangent is one of the primary trigonometric functions. It relates the two sides of a right triangle in relation to a given angle. The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side.

Mathematically, cotangent is expressed as:
  • \( \cot \theta = \frac{adjacent}{opposite} \)
  • Alternatively, as a reciprocal: \( \cot \theta = \frac{1}{\tan \theta} \)
Thus, knowing the cotangent of an angle allows you to easily determine the tangent, and vice versa. This makes it easy to switch between perspective angles or viewpoints in problems involving right triangles.
Tangent
The tangent function is essential in the field of trigonometry. It helps describe the relationship between the sides of a right triangle. Specifically, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Its formula is:
  • \( \tan \theta = \frac{opposite}{adjacent} \)
Tangent is one of the three primary trigonometric ratios, and knowing its value can help determine unknown sides or angles in various trigonometric problems. Understanding the tangent function is crucial for calculating other trigonometric functions such as sin, cos, and their reciprocals.
Scientific Calculator
A scientific calculator is a vital tool for solving trigonometric problems efficiently. These calculators can compute various trigonometric functions, including tangent and cotangent, by performing complex calculations with precision.

Steps to use a scientific calculator for trigonometry include:
  • Ensure your calculator is set to degree mode, not radians, unless required otherwise.
  • Enter the tangent function by using functions often labeled as \( \tan \).
  • Calculate cotangent using the reciprocal function, \( \cot \theta = \frac{1}{\tan \theta} \), sometimes labeled as a reciprocal or \( 1/x \).
By mastering the use of a scientific calculator, you can speed up solving exercises, ensuring accuracy and confirming results through reverse calculations.

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Most popular questions from this chapter

Use the drawings where provided to solve each problem. Angle measures should be given to the nearest degree; distances should be given to the nearest lenth of a unit. The pitch or slope of a roofline is 5 to \(12 .\) Find the measure of angle \(\alpha\). (GRAPH CAN'T COPY)

In Exercises 3 and \(4,\) state the form of the Law of Sines used to solve the problem. Give the answer in a form like \(\frac{\sin 72^{\circ}}{6.3}=\frac{\sin 55^{\circ}}{a}\) a) Find \(\beta\) if it is known that \(a=5\) \(b=8,\) and \(\alpha=40^{\circ}\) b) Find \(c\) if it is known that \(a=5.3, \alpha=41^{\circ}\) and \(\gamma=87^{\circ}\) (GRAPH CANT COPY)

Use either Table 11.2 or a calculator to find the sine of the indicated angle to four decimal places. $$ \sin 46^{\circ}$$

Use a calculator and reciprocal relationships to find each ratio correct to four decimal places. $$\cot 34^{\circ}$$

Angle measures should be given to the nearest degree; distances should be given to the nearest tenth of a unit. When her airplane is descending to land, the pilot notes an angle of depression of \(5^{\circ} .\) If the altimeter shows an altitude reading of \(120 \mathrm{ft}\), what is the distance \(x\) from the plane to touchdown? GRAPH CAN'T COPY.
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