Problem 4 - Precalculus with Limits

Question

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \dfrac{\cos u}{\sin u} \) = ________

Step-by-Step Solution

Answer

The blank should be filled with cot \(u\).

Step 1: Identify the trigonometric function

The given expression is \( \dfrac{\cos u}{\sin u} \). This form is similar to an identity of trigonometric functions which is cotangent function. The cotangent of angle \(u\), denoted as cot \(u\), is the ratio of the cosine to the sine of that angle.

Step 2: Apply the cotangent identity

The cotangent identity is defined as cot \(u\)= \( \dfrac{\cos u}{\sin u} \). This can replace the given expression.

Step 3: Complete the trigonometric identity

Therefore, the blank should be filled with cot \(u\).

Key Concepts

Cotangent

The cotangent function is one of the six fundamental trigonometric ratios. It's often abbreviated as "cot" and used in trigonometry to relate the angles and sides of right-angled triangles. Looking closely at the cotangent function, it is defined as the reciprocal of the tangent function. This means that:

• The cotangent of an angle, denoted as \( \cot u \), is equal to the ratio of the cosine of the angle to the sine of the angle: \( \cot u = \dfrac{\cos u}{\sin u} \).

It's essential to recognize this identity because it simplifies many trigonometric problems. When you encounter an expression like \( \dfrac{\cos u}{\sin u} \), replacing it with \( \cot u \) can make calculations easier, especially when solving equations or verifying identities.

Cosine

Cosine is a key trigonometric function that describes the ratio of the adjacent side to the hypotenuse in right-angled triangles. It's represented by the symbol \( \cos \). When you consider the unit circle, the cosine of an angle corresponds to the x-coordinate of a point on the circle. Here are a few important properties of cosine:

• Cosine is an even function. This means that \( \cos(-u) = \cos u \).
• It has a periodicity of \( 2\pi \), meaning that \( \cos(u + 2\pi) = \cos u \).
• The range of cosine is from -1 to 1, similar to the sine function.

Understanding these properties helps in comprehending trigonometric identities and transformations. For instance, the identity \( \cot u = \dfrac{\cos u}{\sin u} \) makes use of cosine when expressed in terms of cotangent.

Sine

The sine function, symbolized as \( \sin \), is another foundational trigonometric function. It represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. In terms of the unit circle, the sine of an angle corresponds to the y-coordinate of a point on the circle. To grasp sine better, here are some key points:

• Sine is an odd function, meaning \( \sin(-u) = -\sin u \).
• Like cosine, sine also has a periodicity of \( 2\pi \): \( \sin(u + 2\pi) = \sin u \).
• The range of sine values is between -1 and 1.

Recognizing these properties enables us to use sine effectively in various trigonometric identities, such as in the cotangent identity \( \cot u = \dfrac{\cos u}{\sin u} \), highlighting the interplay between sine, cosine, and other trigonometric functions.