Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\cot \theta \sin \theta$$

Step-by-Step Solution

Answer

The simplified expression is \( \cos \theta \).

Step 1: Rewrite Cotangent in Terms of Sine and Cosine

The cotangent function, \( \cot \theta \), can be rewritten in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute this into the original expression: \( \cot \theta \sin \theta = \frac{\cos \theta}{\sin \theta} \times \sin \theta \).

Step 2: Simplify the Expression

In the expression \( \frac{\cos \theta}{\sin \theta} \times \sin \theta \), the \( \sin \theta \) in the numerator and denominator cancel each other out, leaving you with \( \cos \theta \). Thus, the simplified expression in terms of sine and cosine is \( \cos \theta \).

Key Concepts

Cotangent

The cotangent function is one of the fundamental trigonometric functions. It is often abbreviated as "cot" and represents the reciprocal of the tangent function. In the context of a right triangle:

• Tangent (\( \tan \theta \)) is defined as the ratio of the opposite side to the adjacent side.
• Cotangent (\( \cot \theta \)) is the reciprocal of tangent, given by the ratio of the adjacent side to the opposite side.

For any angle \( \theta \), the cotangent can be expressed in terms of sine and cosine, which are more commonly studied trigonometric functions:

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

When used in expressions, like the one we are discussing, substituting cotangent with its sine and cosine form can make simplification more straightforward, as seen in the problem where \( \cot \theta \sin \theta \) was reduced to simply \( \cos \theta \).

Sine

The sine function is an essential part of trigonometry. It is often abbreviated as "sin." This function helps define the relationship between the angles and lengths in right-angled triangles.

• In a right triangle, sine (\( \sin \theta \)) is the ratio of the length of the side opposite to the angle \( \theta \) to the length of the hypotenuse.

This function is also closely aligned with the unit circle, where for an angle \( \theta \):

• The sine of \( \theta \) is the y-coordinate of the point on the unit circle.

Understanding sine is crucial when dealing with trigonometric identities and transformations. It coincides with another core trigonometric function - cosine. We often use sine in combination with other functions, like cosine or cotangent, to simplify trigonometric expressions, as demonstrated in the given problem.

Cosine

Cosine is another cornerstone of trigonometry, represented as "cos." Like sine, it plays a distinct role in understanding the geometry of triangles and the properties of functions.

• In a right triangle, cosine (\( \cos \theta \)) is the ratio of the adjacent side's length to that of the hypotenuse with respect to an angle \( \theta \).

On the unit circle, cosine represents the x-coordinate of the point corresponding to the angle \( \theta \).

• It is periodic, similar to sine, with a period of \( 2\pi \) radians, making it repeat its values with this interval.
• Cosine is particularly handy in expressing various trigonometric relations, like the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).

In the given exercise, transforming the expression \( \cot \theta \sin \theta \) to \( \cos \theta \) illustrates the simplicity and elegance often found within trigonometric transformations and identity usage.