Problem 38
Question
In Exercises \(35-38,\) use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. $$8 \sin ^{2} x \cos ^{2} x$$
Step-by-Step Solution
Verified Answer
The simplified form of \(8 \sin ^{2} x \cos ^{2} x\) using the power-reducing formulas is \(4 \sin ^2x\).
1Step 1: Utilize Power-Reducing Formulas
Begin by substituting the powers of sine and cosine in the given expression \(8 \sin ^{2} x \cos ^{2} x \) with the power-reducing formulas. Therefore, the expression becomes \(8 *\frac{1 - \cos (2x)}{2} * \frac{1 + \cos (2x)}{2}\)
2Step 2: Simplify the Expression
Now, apply the distributive property to the expression \(8 *\frac{1 - \cos (2x)}{2} * \frac{1 + \cos (2x)}{2} \) which equals \(2*(1 - \cos^{2}(2x)) \)
3Step 3: Apply Power-Reducing Formula Again
For the new resulted expression \(2*(1 - \cos^{2}(2x)) \), apply the power-reducing formula of cosine again. Hence, it changes to \(2*(2*\sin^2x) = 4 \sin ^2x\)
Key Concepts
Trigonometric FunctionsTrigonometry in PrecalculusSimplifying Trigonometric Expressions
Trigonometric Functions
Trigonometric functions are the basis of trigonometry, a branch of mathematics concerning the relationships between the angles and sides of triangles. The most commonly used trigonometric functions are sine (sin), cosine (cos), and tangent (tan), along with their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).
These functions are fundamental in studying waves, oscillations, and many other applications including engineering, physics, and even in finance. Understanding how to manipulate these functions through various trigonometric identities, like power-reducing formulas, is essential in simplifying complex expressions and solving equations.
The power-reducing formulas are particularly useful when you need to convert higher powers of sine and cosine into expressions with a power of one or less. These formulas are derived from the double-angle formulas and the Pythagorean identity. A mastery of these formulas allows one to approach problems in precalculus and calculus with greater confidence and simplicity.
These functions are fundamental in studying waves, oscillations, and many other applications including engineering, physics, and even in finance. Understanding how to manipulate these functions through various trigonometric identities, like power-reducing formulas, is essential in simplifying complex expressions and solving equations.
The power-reducing formulas are particularly useful when you need to convert higher powers of sine and cosine into expressions with a power of one or less. These formulas are derived from the double-angle formulas and the Pythagorean identity. A mastery of these formulas allows one to approach problems in precalculus and calculus with greater confidence and simplicity.
Trigonometry in Precalculus
In precalculus, trigonometry serves as a critical component that paves the way for calculus concepts. It connects the geometric world with the algebraic world and provides tools for solving problems involving triangles and circles.
Students are introduced to the unit circle, which is a circle with a radius of one centered at the origin of a coordinate system. The unit circle is a visual aide for understanding angles and the associated values of trigonometric functions at those angles. Further, it plays a key role in analyzing periodic functions and harmonic motion.
The study of trigonometry in precalculus is diverse, covering topics like trigonometric identities, solving trigonometric equations, and graphing trigonometric functions. Grasping these concepts is vital to succeed not just in precalculus, but also in the higher-level courses that follow. Power-reducing formulas are one of the many tools that students will encounter, which help in simplifying expressions and solving equations efficiently.
Students are introduced to the unit circle, which is a circle with a radius of one centered at the origin of a coordinate system. The unit circle is a visual aide for understanding angles and the associated values of trigonometric functions at those angles. Further, it plays a key role in analyzing periodic functions and harmonic motion.
The study of trigonometry in precalculus is diverse, covering topics like trigonometric identities, solving trigonometric equations, and graphing trigonometric functions. Grasping these concepts is vital to succeed not just in precalculus, but also in the higher-level courses that follow. Power-reducing formulas are one of the many tools that students will encounter, which help in simplifying expressions and solving equations efficiently.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions can often require a bit of algebraic finesse and a good grasp of trigonometric identities. Power-reducing formulas are incredibly useful identities that help in this simplification process.
These formulas express higher powers of sine and cosine functions in terms of the first power. For instance, the power-reducing formula for \( \sin^{2}(x) \) is \( \frac{1 - \cos(2x)}{2} \) and for \( \cos^{2}(x) \) is \( \frac{1 + \cos(2x)}{2} \)\. When faced with expressions like \( 8 \sin^{2}(x) \cos^{2}(x) \) we can apply these formulas to reduce the expression into a simpler form that is easier to evaluate or further manipulate.
Using these formulas not only helps in finding solutions more readily but also in understanding the underlying properties of trigonometric functions. This understanding is crucial, particularly as the mathematics becomes more complex. It ensures students are better prepared to tackle advanced problems, whether in calculus, physics, or beyond.
These formulas express higher powers of sine and cosine functions in terms of the first power. For instance, the power-reducing formula for \( \sin^{2}(x) \) is \( \frac{1 - \cos(2x)}{2} \) and for \( \cos^{2}(x) \) is \( \frac{1 + \cos(2x)}{2} \)\. When faced with expressions like \( 8 \sin^{2}(x) \cos^{2}(x) \) we can apply these formulas to reduce the expression into a simpler form that is easier to evaluate or further manipulate.
Using these formulas not only helps in finding solutions more readily but also in understanding the underlying properties of trigonometric functions. This understanding is crucial, particularly as the mathematics becomes more complex. It ensures students are better prepared to tackle advanced problems, whether in calculus, physics, or beyond.
Other exercises in this chapter
Problem 38
Verify each identity. $$\frac{\tan ^{2} x-\cot ^{2} x}{\tan x+\cot x}=\tan x-\cot x$$
View solution Problem 38
Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\sin \left(2 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$$
View solution Problem 38
Verify each identity. $$\tan (\pi-x)=-\tan x$$
View solution Problem 39
Use words to describe the given formula. $$\sin \alpha \sin \beta=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)]$$
View solution