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Chapter 1: Problem 96

Solve each problem. If \(\tan \alpha=1 / 70,\) then what is \(\cot \alpha ?\)

Short Answer

Expert verified
\tan \alpha \ =70.

Step by step solution

01

Identify Given Information

Given that \(\tan \ \alpha = \ \ \frac{1}{70}\), we need to find \(\tan\) and \(\alpha\).
02

Reciprocal Identity

We know the reciprocal identity of tangent: \(\tan\ \alpha\) is the reciprocal of \(\cot\) and vice versa. So: \(\tan \alpha = \ \ \frac{1}{70} \).

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Key Concepts

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

Reciprocal Identities
Reciprocal identities are fundamental in understanding trigonometric functions. When we talk about reciprocals in math, we mean that two numbers multiply together to make 1. For instance, the reciprocal of 2 is \frac{1}{2} because \[ 2 \times \frac{1}{2} = 1.\] This concept is widely used in trigonometry. The reciprocal identities for trigonometric functions link pairs of functions. For example:
  • \(\tan \theta = \frac{1}{\textrm{cot} \theta}\)
  • \(\textrm{cot} \theta = \frac{1}{\tan \theta}\)
These identities help simplify problems and find unknown values when dealing with trigonometric equations. If you know \(\tan \theta\), finding \(\textrm{cot} \theta\) becomes straightforward through their reciprocal relationship.
Tangent
The tangent function \(\tan\) is one of the basic trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In mathematical terms: \[ \tan \theta = \frac{\textrm{opposite}}{\textrm{adjacent}} \] Tangent can also be related to other trigonometric functions: \[ \tan \theta = \frac{\textrm{sin} \theta}{\textrm{cos} \theta} \] Knowing these relationships helps us understand and solve trigonometric problems. For example, if it is given that \(\tan \alpha = \frac{1}{70}\) \, then using the reciprocal identities makes it easy to find \(\textrm{cot} \alpha\)
Cotangent
The cotangent function \(\textrm{cot} \theta\) is another fundamental trigonometric function. It is the reciprocal of the tangent function. In other words: \[ \textrm{cot} \theta = \frac{1}{\tan \theta} \] Cotangent can also be defined as the ratio of the adjacent side to the opposite side in a right-angled triangle: \[ \textrm{cot} \theta = \frac{\textrm{adjacent}}{\textrm{opposite}} \] For our given problem, we know \(\tan \alpha = \frac{1}{70} \). To find \(\textrm{cot} \alpha\), we use the reciprocal identity: \[ \textrm{cot} \alpha = \frac{1}{\tan \alpha} = \frac{1}{\frac{1}{70}} = 70. \] So, \(\textrm{cot} \alpha = 70\).

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