Problem 83 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the first section of the next chapter. The exercises use identities, introduced in Section 5.2, that enable you to rewrite trigonometric expressions so that they contain only sines and cosines: $$ \begin{aligned} \csc x &=\frac{1}{\sin x} & & \sec x=\frac{1}{\cos x} \\ \tan x &=\frac{\sin x}{\cos x} & & \cot x=\frac{\cos x}{\sin x} \end{aligned} $$ Rewrite each expression by changing to sines and cosines. Then simplify the resulting expression. $$ \tan x \csc x \cos x $$

Step-by-Step Solution

Answer

The simplified form of expression \( \tan x \csc x \cos x \) is '1'

Step 1: Substitute given trigonometric identities

Replace every trigonometric function in the expression with related identity in terms of sines and cosines. For \( \tan x \), \( \csc x \) and \( \cos x \), substitute their respective identities, i.e., \( \frac{\sin x}{\cos x} \), \( \frac{1}{\sin x} \) and \( \cos x \) respectively.\nSo, \( \tan x \csc x \cos x \) will turn out to be \( \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} \cdot \cos x \)

Step 2: Simplify the expression

Simplify the resulting expression by cancelling common terms. Here, \( \sin x \) in numerator and denominator can be cancelled as well as \( \cos x \) in numerator and denominator can be cancelled. On performing these operations, the resulting expression is '1'

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships with the sides of a triangle. In this context, these functions help describe circular and oscillatory motion. The primary trigonometric functions include:

• Sine (\( \sin x \)): Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos x \)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan x \)): Is the ratio of the sine and cosine functions, essentially opposite over adjacent.
• Cosecant (\( \csc x \)): Is the reciprocal of sine, or hypotenuse over opposite.
• Secant (\( \sec x \)): Is the reciprocal of cosine, or hypotenuse over adjacent.
• Cotangent (\( \cot x \)): Is the reciprocal of tangent, or adjacent over opposite.

These functions are interconnected and can be expressed in terms of each other using identities. This interdependence allows for simplifying trigonometric expressions, an essential skill in math and physics.

Sine and Cosine

The sine and cosine functions are perhaps the most pivotal trigonometric functions. They form the basis for other functions like tangent, secant, cosecant, and cotangent. Understanding how to manipulate these functions is crucial:

• The function \( \sin x \) describes a curve depicting a cycle which starts from zero, peaks at one, descends back to zero, goes to negative one, and returns to zero.
• Similarly, \( \cos x \) describes a curve that begins from one, falls to zero, goes to negative one, returns to zero, and rises back to one.

These cycles are essential when considering periodic phenomena, such as oscillations and sound waves. Furthermore, any trigonometric expression can often be rewritten and simplified using only sine and cosine because:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

Developing fluency in these identities supports simplifying complex trigonometric equations, modeling real-world phenomena, and solving problems in calculus and physics.

Simplifying Expressions

Simplifying expressions with trigonometric identities is a valuable technique that enhances solving capacities in mathematics. By restating an expression with fundamental identities, one can make the expression easier to handle.
For example, let's examine the expression \( \tan x \csc x \cos x \). To simplify, we substitute each function with its sine and cosine equivalent:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)
• \( \cos x = \cos x \)

Putting it all together, we have:\[\frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} \cdot \cos x\]Now, simplify by cancelling out common terms:

• \( \sin x \) cancels with \( \sin x \) in the numerator and denominator.
• \( \cos x \) cancels with \( \cos x \) in the numerator and denominator.

After cancellation, what remains is simply '1'.
This simplification showcases the power of using fundamental identities effectively. Understanding and applying these techniques fluently provides significant advantages in solving trigonometric problems.