Problem 84 - 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. $$ \sec x+\tan x $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sec x+\tan x\) in terms of sines and cosines is \(\frac{1+\sin x}{\cos x}\)

Step 1: Substitute Identities

Replace the given trigonometric identities \(\sec x\) and \(\tan x\) with their equivalent forms in terms of sines and cosines. So, \(\sec x\) becomes \(\frac{1}{\cos x}\) and \(\tan x\) becomes \(\frac{\sin x}{\cos x}\)

Step 2: Add Expressions

Add together these expressions, \(\frac{1}{\cos x}\) + \(\frac{\sin x}{\cos x}\)

Step 3: Simplify the Expression

Because these two fractions have the same denominator \(\cos x\), they can be added together more easily. The resulting expression is \(\frac{1+\sin x}{\cos x}\)

Key Concepts

Sine and Cosine Conversion

When dealing with trigonometric expressions, simplifying them into terms of sine \((\sin x)\) and cosine \((\cos x)\) can make the equations easier to manage.This process is known as sine and cosine conversion.By rewriting the functions in this way, you can simplify complex expressions and perform further operations smoothly.

For instance, functions such as secant \((\sec x)\) and tangent \((\tan x)\), can be converted into equations utilizing sine and cosine values:

• The secant, \(\sec x\), is rewritten as \(\frac{1}{\cos x}\).
• The tangent, \(\tan x\), turns into \(\frac{\sin x}{\cos x}\).

The key advantage of converting into sines and cosines is the uniformity it brings to the trigonometric functions. This makes computation easier and aids in identifying simplifications or recognizing certain patterns in expressions.

Trigonometric Expressions

Trigonometric expressions often seem complicated at first glance, but with an understanding of key identities, they become easier to handle.You aim to change functions like secant and tangent into their trigonometric base counterparts to streamline your calculations.Using trigonometric identities helps in rewriting any complex trigonometric expressions into simpler forms.

For example, in the given expression, \(\sec x + \tan x\),we used their identities to turn them into trigonometric expressions composed of sine and cosine:

• \(\sec x\) becomes \(\frac{1}{\cos x}\)
• \(\tan x\) converts to \(\frac{\sin x}{\cos x}\)

After rewriting these in terms of sine and cosine, you can easily perform operations like addition or subtraction to simplify the expression further.

This process allows for a structured approach to solving and understanding trigonometric expressions, leading to mastery over them beyond just solving for specific values.

Secant and Tangent Identities

Understanding the identities and conversions of secant and tangent forms an essential part of simplifying trigonometric equations.Knowing these identities helps not only in simplifying equations but also in solving equations in broader mathematical problems.

The secant identity is particularly important as \(\sec x\) stands for \(\frac{1}{\cos x}\), which represents the reciprocal of cosine.Similarly, the tangent identity, \(\tan x = \frac{\sin x}{\cos x}\), expresses it as a ratio of sine to cosine.

These identities are instrumental when dealing with trigonometric equations where simplification or conversion of forms is necessary. By applying these identities, you're often able to break down complex problems into easier parts. This not only aids in problem-solving but also builds a deeper comprehension of the relationships between different trigonometric functions.