Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \((\sin \theta)(\cot \theta)\)

Step-by-Step Solution

Answer

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

Step 1: Write Cotangent in Terms of Sine and Cosine

We know that the cotangent function can be expressed in terms of sine and cosine. Specifically, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Step 2: Substitute into the Original Expression

Replace \(\cot \theta\) in the expression \((\sin \theta)(\cot \theta)\) with \(\frac{\cos \theta}{\sin \theta}\). This gives us \((\sin \theta)\left(\frac{\cos \theta}{\sin \theta}\right)\).

Step 3: Simplify the Expression

When we multiply the terms, the \(\sin \theta\) in the numerator and denominator will cancel each other out. Thus, the expression simplifies to \(\cos \theta\).

Key Concepts

Sine and Cosine

Sine and cosine are two of the most fundamental trigonometric functions. They are crucial in understanding the relationships within right-angled triangles.

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Mathematically, it is expressed as \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}.\)
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. It is expressed as \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}.\)

These functions are periodic, repeating their values in regular intervals of \(360^\circ\) or \(2\pi\) radians. In the unit circle, the sine of an angle corresponds to the \(y\)-coordinate, and the cosine corresponds to the \(x\)-coordinate of a point on the circle. This geometric approach makes it easier to visualize their behavior.
Knowing these basic relationships can simplify more complex expressions into something more manageable using known identities.

Cotangent

The cotangent function is another trigonometric function derived from sine and cosine. It is the reciprocal of the tangent function and provides a measure for the ratio between the adjacent and opposite side of a right-angled triangle.To express cotangent in terms of sine and cosine:

• The tangent function is defined as the ratio of sine over cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}).\)
• Thus, the cotangent \(\cot \theta\) is the inverse of tangent: \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}).\)

This identity is useful when expressing trigonometric functions in terms of sine and cosine, simplifying equations, and sometimes diagnosing fundamental trigonometric transformations.
By swapping out tangent for sine and cosine components through cotangent, the process of simplification often becomes more intuitive and straightforward.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill in mathematics, enabling complex problems to become solvable through simplification. Understanding the identities of trigonometric functions like sine, cosine, and cotangent allows us to rewrite expressions efficiently.Start by recognizing the identities involved. For instance:

• A common simplification involves canceling terms such as in the expression \((\sin \theta)(\cot \theta)\).
  By substituting \(\cot \theta\) with \(\frac{\cos \theta}{\sin \theta}\), we can see: \(\sin \theta \cdot \frac{\cos \theta}{\sin \theta} = \cos \theta).\)
• The \(\sin \theta\) terms cancel, allowing focus on remaining functions (e.g., \(\cos \theta\)).

This method often involves deriving simpler expressions, which are easier to evaluate or integrate in further mathematical contexts.
Mastering these techniques leads to gaining deeper insights and efficiently managing more complicated mathematical operations.