Problem 50 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t) /(\tan t)$$

Step-by-Step Solution

Answer

Question: Simplify the expression \((\sin t) /(\tan t)\). Answer: \(\cos t\)

Step 1: Write down the given expression

Here is the given expression. $$\frac{\sin t}{\tan t}$$

Step 2: Replace tan t with its identity

Recall that \(\tan t = \frac{\sin t}{\cos t}\). Substitute this into the given expression, replacing \(\tan t\) with \(\frac{\sin t}{\cos t}\). $$\frac{\sin t}{\frac{\sin t}{\cos t}}$$

Step 3: Simplify the expression

To simplify, we multiply the numerator and denominator by the reciprocal of the denominator. $$\frac{\sin t}{\frac{\sin t}{\cos t}} \cdot \frac{\cos t}{\sin t}$$ Then, cancel out the common terms (\(\sin t\)) in the numerator and denominator. $$\frac{\cancel{\sin t}}{\cancel{\sin t}} \cdot \cos t = \cos t$$

Step 4: Write down the final simplified expression

After simplifying, we find that the original expression \((\sin t) /(\tan t)\) is equal to \(\cos t\). $$\frac{\sin t}{\tan t} \:=\: \cos t$$

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of making an expression easier to understand or work with, by reducing it to its simplest form. When simplifying algebraic expressions, such as trigonometric ones, we often use known identities or rules to transform the expressions into more manageable forms.

To simplify an expression like \(\frac{\sin t}{\tan t}\), you should start by identifying any trigonometric identities that you can apply. For example, the identity \(\tan t = \frac{\sin t}{\cos t}\) allows us to substitute \(\tan t\) with \(\frac{\sin t}{\cos t}\).

• This substitution is crucial because it converts the expression into a form where simple algebraic operations, such as division, can be easily applied.
• Furthermore, by applying basic algebra, we can simplify the expression further by canceling like terms. This helps us understand the essential part of the trigonometric expression.

Using algebraic simplification not only helps solve the mathematical problem at hand but also paves the way toward deeper understanding of how various mathematical rules interplay.

Trigonometric Functions

Trigonometric functions are fundamental to trigonometry and are used to describe angles and solve problems involving triangles. These functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), amongst others. Here’s why they are important:

• The sine function, \(\sin t\), represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• The cosine function, \(\cos t\), is the ratio of the adjacent side to the hypotenuse.
• The tangent function, \(\tan t\), combines these two, being the ratio of \(\sin t\) to \(\cos t\).

In practice,
trigonometric functions help us manipulate and simplify expressions. For instance, by knowing that \(\tan t = \frac{\sin t}{\cos t}\), you can transform the given problem \(\frac{\sin t}{\tan t}\) into a simpler form where the \(\sin t\) terms can be canceled, revealing the expression's simpler equivalent, \(\cos t\).

Understanding these relationships and rules makes it easier to work with angles and solve complex mathematical problems efficiently.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition and multiplication) that together represent a value. They can also include complex numbers, algebraic terms, and functions, such as trigonometric functions.

• An expression provides a way to represent relationships between different mathematical elements.
• Expressions can often be simplified using known mathematical identities and rules.

For the expression \(\frac{\sin t}{\tan t}\), the goal is to simplify it using trigonometric identities. Continuing from our previous steps, the known identity \(\tan t = \frac{\sin t}{\cos t}\) allows us to replace \(\tan t\) with this expression, resulting in a chance to apply algebraic simplification.

Understanding how to handle various mathematical expressions, apply identities, and simplify them is crucial in problem-solving. This not only helps reveal the fundamental relationships within the equations but also aids in presenting clear solutions.