Question

Change to an expression containing only sin and cos. $$\cot x+\csc x$$

Short answer

\(\cot x + \csc x = \frac{\cos x}{\sin x} + \frac{1}{\sin x} = \frac{\cos x + 1}{\sin x}\)

Step-by-step solution

Convert \(\cot x\) to \(\sin x\) and \(\cos x\)

Recall that \(\cot x = \frac{\cos x}{\sin x}\). So, replace \(\cot x\) with \(\frac{\cos x}{\sin x}\).

Convert \(\csc x\) to \(\sin x\) and \(\cos x\)

Recall that \(\csc x = \frac{1}{\sin x}\). So, replace \(\csc x\) with \(\frac{1}{\sin x}\).

Combine the terms

Now, combine the expressions \(\frac{\cos x}{\sin x}\) and \(\frac{1}{\sin x}\) to obtain a single expression in terms of \(\sin x\) and \(\cos x\).

Key concepts

Cotangent to Sine and Cosine

Understanding how to convert cotangent into sine and cosine is vital for simplifying trigonometric expressions. The cotangent of an angle, often represented as \( \cot x \), is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, but can also be defined in terms of the sine \( \sin x \) and cosine \( \cos x \) functions as the reciprocal of the tangent: \( \cot x = \frac{1}{\tan x}\). Since \(\tan x = \frac{\sin x}{\cos x}\), cotangent can be expressed as \( \cot x = \frac{\cos x}{\sin x}\).

Remember that \(\cos x\) corresponds to the adjacent side of the triangle and \(\sin x\) to the opposite side, relative to the angle in consideration. By expressing cotangent in terms of \(\sin x\) and \(\cos x\), it allows for simplification in complex trigonometric problems and can aid in the computation of exact values when the sine or cosine of the angle is known, or when solving equations.

Cosecant to Sine

Moving on to the cosecant function, it's a less commonly used trigonometric function but still important in the grand scheme of trigonometric identities. The cosecant of an angle, denoted by \( \csc x \), is the reciprocal of the sine function, making its relationship straightforward: \( \csc x = \frac{1}{\sin x}\).

Framing cosecant in terms of sine is beneficial because sine is one of the fundamental trigonometric functions, and many theorems, properties, and calculations in trigonometry are described using sine. Converting \( \csc x \) to \( \sin x \) provides a way to simplify expressions and enables us to integrate the term into more complex equations where the sine function is required for solving or simplification.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill in mathematics, enabling students to work more efficiently with trigonometric equations and functions. The simplification process often involves applying trigonometric identities to rewrite expressions in a more manageable form. For example, start by identifying opportunities to convert functions like cotangent and cosecant to sine and cosine, as seen in the exercise conversion of \(\cot x + \csc x\).

Once expressions are in terms of sine and cosine, they can often be combined or further simplified due to the properties of these fundamental functions. As in our exercise, after converting cotangent and cosecant to the sine function, they can be easily added because they now have a common denominator. Moreover, students should look for chances to factor expressions, use Pythagorean identities, or even perform function composition as a way to simplify trigonometric expressions.

Practice Tip: It's always a good idea to familiarize oneself with the various trigonometric identities, as these will serve as the tools for simplification. With practice, recognizing when and how to apply these identities will become second nature, making the simplification process quicker and more intuitive.