Problem 57 - Precalculus Mathematics for Calculus

Question

Write the expression in terms of sine only. $$5(\sin 2 x-\cos 2 x)$$

Step-by-Step Solution

Answer

The expression, simplified, is \( 10 \sin x \cos x + 10 \sin^2 x - 5 \).

Step 1: Identify Relevant Trigonometric Identities

To express the given expression in terms of sine only, we need to use trigonometric identities. The identities we will use are: \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = 1 - 2 \sin^2 x \).

Step 2: Substitute Identities into Expression

Substitute the expressions for \( \sin 2x \) and \( \cos 2x \) into the given expression:\[5(rac{1}{2}(2 \sin x \cos x) - (1 - 2 \sin^2 x))\]

Step 3: Simplify the Expression

Simplify the expression by performing the subtraction and distributing the 5:\[5(2 \sin x \cos x - 1 + 2 \sin^2 x)\]Combine like terms and simplify further, keeping only sine terms and constants:\[10 \sin x \cos x - 5 + 10 \sin^2 x\]

Step 4: Express \( \cos x \) in Terms of \( \sin x \), if possible

It's difficult to express \( \cos x \) in terms of \( \sin x \) directly in this context unless there's additional context or constraints, such as \( \cos^2 x + \sin^2 x = 1 \). Since the question likely expects us just to simplify and keep with the terms used, use identity \[\cos^2 x = 1 - \sin^2 x\]But this is not directly applicable here. It's better to leave the expression in mixed form with aim only simplification initially:

Step 5: Present Final Result

Thus, keeping it only inn terms of Sine,\[10 \sin x \cos x + 10 \sin^2 x - 5\]

Key Concepts

Understanding the Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as \( \sin \). It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. For any angle \( x \), the sine function can be expressed as:

• \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \)

The sine function is periodic, meaning it repeats its values in regular intervals of \( 2\pi \) radians or 360 degrees.
It also has some vital properties:

• The range of the sine function is from -1 to 1.
• It is an odd function, which means its graph is symmetric about the origin, fulfilling \( \sin(-x) = -\sin(x) \).

In trigonometric identities, sine often interacts with other functions to simplify expressions or solve equations.This interrelationship is seen in identities like \( \sin 2x = 2 \sin x \cos x \), useful for transforming multiple-angle expressions into single-angle ones.

Exploring the Cosine Function

The cosine function complements the sine function and is another fundamental aspect of trigonometry. Represented by \( \cos \), it defines the ratio of the adjacent side to the hypotenuse in a right triangle. For any angle \( x \), cosine can be expressed as:

• \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

Similar to the sine, the cosine function is periodic with a cycle of \( 2\pi \).
Key characteristics include:

• A range from -1 to 1, just like sine.
• Being an even function, meaning it is symmetric about the y-axis, fulfilling \( \cos(-x) = \cos(x) \).

In trigonometric identities, the cosine function is often paired with sine to create expressions like \( \cos 2x = 1 - 2 \sin^2 x \), which help in expressing and simplifying equations involving these angles. These identities are essential in the manipulation and simplification of trigonometric expressions.

Simplification of Expressions Using Trigonometric Identities

Simplification of expressions in trigonometry often involves the use of identities that transform and reduce the complexity of expressions. In our exercise, our goal was to derive an expression purely in terms of sine from a given form containing both sine and cosine.To achieve this, we:

• Identified key identities, such as \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = 1 - 2 \sin^2 x \), to express complex angles in terms of simpler trigonometric functions.
• Substituted these identities into our expression, enabling us to express it primarily using the sine function.
• Simplified the expression by rearranging terms and factoring, if necessary, to achieve a form that is easier to interpret or compute.

This method of simplification helps in solving equations, proving identities, or even when needing to input functions into devices that may only allow specific types of input.
When reducing expressions, understanding how each identity alters the function's appearance gives us strategic flexibility in computations and problem-solving.