Problem 49 - Contemporary Precalculus

Question

In Exercises \(49-54\), prove the given identity. $$\tan t=\frac{1}{\cot t}[\text {Hint}: \text { See page } 497]$$

Step-by-Step Solution

Answer

Question: Prove that the given identity \(\tan t=\frac{1}{\cot t}\) is true. Answer: We proved the given identity by first recalling the definitions of the tangent and cotangent functions, and then relating the tangent function to the cotangent function. By substituting the expression for cotangent and simplifying, we showed that the given identity is true.

Step 1: Recall the definitions of \(\tan\) and \(\cot\)

The tangent function, \(\tan\), is defined as the ratio of the sine function to the cosine function, i.e, $$\tan t = \frac{\sin t}{\cos t}$$ The cotangent function, \(\cot\), is defined as the ratio of the cosine function to the sine function, i.e, $$\cot t = \frac{\cos t}{\sin t}$$

Step 2: Relate the \(\tan\) function to the \(\cot\) function

As we know that the \(\cot t\) is the reciprocal of the \(\tan t\), we can rewrite the \(\cot t\) expression as follows: $$\cot t = \frac{1}{\tan t}$$ Substitute the definition of the tangent function: $$\cot t = \frac{1}{\dfrac{\sin t}{\cos t}}$$

Step 3: Substitute the expression for \(\cot t\) and simplify

Now, we will substitute the expression of \(\cot t\) into the given identity and simplify it: $$\tan t = \frac{1}{\cot t} = \frac{1}{\dfrac{1}{\dfrac{\sin t}{\cos t}}}$$ Simplify it by taking the reciprocal of \(\cot t\): $$\tan t = \frac{\sin t}{\cos t}$$

Step 4: Conclusion

We have shown that the given identity, $$\tan t = \frac{1}{\cot t}$$ is true, as we were able to simplify it to $$\tan t = \frac{\sin t}{\cos t}$$ which is the definition of the tangent function.

Key Concepts

Tangent Function

The tangent function, abbreviated as \( \tan \), is a trigonometric function that arises frequently in mathematics, particularly in the study of triangles and circles. It is defined as the ratio of the sine and cosine functions. Specifically, the tangent of an angle \( t \) is given by: \[ \tan t = \frac{\sin t}{\cos t} \] This means that for any angle, the tangent function will take the value of the sine of that angle divided by the cosine of the same angle.
This relationship helps in understanding and calculating the angle and side length relationships in right triangles.

• In a right triangle, the tangent of an angle represents the ratio of the opposite side to the adjacent side.
• Tangent is periodic with a period of \( \pi \) or 180 degrees, meaning \( \tan(t + \pi) = \tan t \).

Understanding the tangent function is crucial in solving various trigonometric identities and proofs.

Cotangent Function

The cotangent function, represented as \( \cot \), is another key function in trigonometry that is the reciprocal of tangent. If an angle \( t \) is given, the cotangent of that angle is expressed as the cosine of the angle divided by its sine: \[ \cot t = \frac{\cos t}{\sin t} \]
Because it is the reciprocal of the tangent function, there is a simple, useful reciprocal identity:

• \( \cot t = \frac{1}{\tan t} \)

Cotangent appears often in trigonometry problems and can be helpful in solving equations where the tangent function is involved. It is also periodic but with a period of \( \pi \), similar to the tangent function.
In the context of a right triangle, cotangent represents the ratio of the adjacent side to the opposite side. Mastering the cotangent function and its relationship with tangent provides a deeper understanding of trigonometric identities.

Reciprocal Identities

Reciprocal identities are a fundamental concept in trigonometry, forming the backbone of many simplifications and proofs. These identities help link the primary trigonometric functions through their reciprocal relations. For instance, the cotangent and tangent functions are reciprocals: \[ \cot t = \frac{1}{\tan t} \]
This relationship showcases how tangent and cotangent are intertwined. By knowing one, you can instantly infer the other. Reciprocal identities also apply to other trigonometric functions besides tangent and cotangent, such as sine and cosecant, or cosine and secant.
Understanding and using reciprocal identities:

• Simplifies expressions and calculations, making solving complex trigonometric problems more manageable.
• Facilitates converting between different trigonometric functions.

These identities are vital tools in trigonometry proofs, making them indispensable for students aiming to gain a strong grip on the subject.

Trigonometry Proofs

Trigonometry proofs are logical arguments that demonstrate the truth of a trigonometric identity or statement. They are crucial for validating relationships between trigonometric functions. In the given exercise, the statement proved was \( \tan t = \frac{1}{\cot t} \). This proof involves several steps:

• Understanding the definitions of the involved trigonometric functions.
• Expressing these functions in terms of sine and cosine, their most basic forms.
• Employing reciprocal identities to transform one side of the equation to match the other.

Proofs like these are solved by logical reasoning, requiring a strong understanding of trigonometric identities.
To develop skills in trigonometry proofs:

• Practice regularly, focusing on different types of identities.
• Become familiar with common strategies such as substituting identities and simplifying fractions.

Mastering trigonometry proofs contributes to a deeper comprehension of the subject and improves problem-solving skills in mathematics overall.