Problem 4 - Algebra and Trigonometry

Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \cos t \csc t $$

Step-by-Step Solution

Answer

\( \cos t \csc t = \cot t \)

Step 1: Expand Use of Trigonometric Identities

Recall that the cosecant function is the reciprocal of the sine function. Therefore, we can express \( \csc t \) in terms of \( \sin t \): \[ \csc t = \frac{1}{\sin t} \] This gives us: \[ \cos t \csc t = \cos t \times \frac{1}{\sin t} \]

Step 2: Simplify the Expression

Simplify the expression obtained. By multiplying \( \cos t \) with \( \frac{1}{\sin t} \), we have: \[ \frac{\cos t}{\sin t} \]

Step 3: Use Trigonometric Identity to Simplify Further

Recognize that \( \frac{\cos t}{\sin t} \) is the definition of the cotangent function. Hence, we have:\[ \cos t \csc t = \cot t \]

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions commonly used in mathematics. It helps us understand the relationship between the angles and side lengths in right-angled triangles. When you encounter a right triangle, the sine of an angle is calculated by taking the length of the side opposite the angle and dividing it by the length of the hypotenuse, which is the triangle's longest side.In mathematical terms:

• The sine of an angle \( t \), denoted as \( \sin t \), is defined by the ratio \( \frac{\text{opposite side}}{\text{hypotenuse}} \).

This function is periodic, with a regular repeating pattern over its domain, which is all real numbers. It has values ranging between -1 and 1, inclusive.The sine function is crucial in trigonometry and analysis, as it establishes a vital link for expressing other trigonometric functions and equations. Its simplicity underlies various identities and formulas, such as the Pythagorean identity:

• \( \sin^2 t + \cos^2 t = 1 \).

Cosine Function

The cosine function works hand in hand with the sine function, helping to fully describe angles and the lengths of sides in right-angled triangles. While sine uses the opposite side in its ratio, cosine uses the adjacent side, emphasizing the difference in these close but distinct trigonometric concepts.To define it mathematically:

• The cosine of an angle \( t \), symbolized \( \cos t \), represents the ratio \( \frac{\text{adjacent side}}{\text{hypotenuse}} \).

Much like sine, the cosine function is periodic, repeating its values in a set cycle as the angle \( t \) progresses through its range. It also varies between -1 and 1.The cosine function, just as essential as sine, appears in numerous trigonometric identities and formulas. It helps determine the value of other functions such as the tangent and secant, and it plays a part in the Pythagorean identity mentioned previously:

• \( \sin^2 t + \cos^2 t = 1 \).

Cosecant Function

The cosecant function isn't as widely known as sine or cosine, but it has its own place in the trigonometric family. Derived from the sine function, the cosecant (\( \csc \)) offers insight into the reciprocal relationships among trigonometric functions.By definition, the cosecant of an angle \( t \) is:

• \( \csc t = \frac{1}{\sin t} \)

This means it is the reciprocal of the sine function, allowing us to express it in terms of sine. Since the values of the sine function range from -1 to 1, the cosecant function will yield results that are either greater than or equal to 1, or less than or equal to -1, except where the sine function is zero (as \( \csc t \) would be undefined there).Though less commonly used alone, the cosecant function is very useful in transformations and mathematical proofs, contributing to simplifying expressions and solving for unknowns in complex trigonometric equations. It is instrumental when dividing or expressing particular angles in terms of other functions.