Problem 50 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\tan \theta \cot \theta=1$$

Step-by-Step Solution

Answer

The identity is verified: \( \tan \theta \cot \theta = 1 \).

Step 1: Express Tangent and Cotangent in Terms of Sine and Cosine

The tangent of an angle \( \theta \) is given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Similarly, the cotangent of \( \theta \) is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 2: Substitute Expressions into the Identity

Substitute the expressions for \( \tan \theta \) and \( \cot \theta \) into the left side of the identity: \( \tan \theta \cot \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right) \).

Step 3: Simplify the Expression

In the expression \( \left(\frac{\sin \theta}{\cos \theta}\right) \left(\frac{\cos \theta}{\sin \theta}\right) \), both the \( \sin \theta \) and \( \cos \theta \) terms cancel out, resulting in \( \frac{\sin \theta \cos \theta}{\cos \theta \sin \theta} = 1 \).

Step 4: Conclude the Verification

We have transformed the left-hand side into \( 1 \), which is the same as the right-hand side. This verifies the identity \( \tan \theta \cot \theta = 1 \).

Key Concepts

Tangent

In trigonometry, the tangent of an angle \( \theta \) is a fundamental ratio derived from the relationship between the sides of a right triangle. It is defined as the ratio of the opposite side to the adjacent side of the angle. In mathematical terms, it is expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Tangent can also be represented using the sine and cosine functions, which makes it very versatile:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This identity is useful for manipulating and transforming equations. For example, when verifying identities like \( \tan \theta \cot \theta=1 \), expressing tangent in terms of sine and cosine allows a pathway to simplify and verify complex trigonometric expressions. By utilizing this relationship, tangent plays a key role in solving various trigonometric equations, offering students a solid foundation for tackling larger problems in trigonometry.

Cotangent

Cotangent is another essential trigonometric function closely related to tangent. It is defined as the reciprocal of the tangent function, meaning it inverses the ratio. Therefore, the cotangent of an angle \( \theta \) can be expressed as:

• \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)

Connecting cotangent to sine and cosine, we get another clear definition:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In trigonometric identities, the role of cotangent becomes evident when simplifying expressions like \( \tan \theta \cot \theta \). Using their sine and cosine definitions, the product simplifies directly to one:

• \( \tan \theta \cot \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right) = 1 \)

Cotangent assists in balancing trigonometric equations and provides insight into symmetry and transformations within trigonometrical graphs. It is critical for students to understand cotangent, as it frequently appears in calculus and physics problems.

Sine and Cosine

Sine and cosine are the foundational building blocks of trigonometry. They define the primary relationships between angles and side lengths in a right triangle. The sine of an angle \( \theta \) is:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

Whereas the cosine is:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These ratios not only define the trigonometric landscape but also help in expressing other trigonometric functions, including tangent and cotangent. As seen in the original exercise, sine and cosine are pivotal for derivations:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

The sine and cosine functions are cyclic, corresponding to the unit circle's properties, which enable the study and modeling of periodic phenomena such as sound waves and alternating currents. Understanding sine and cosine ensures a deeper grasp of trigonometry, enabling students to explore and solve complex mathematical problems confidently.