Problem 23
Question
Verify the identity. $$4 \cos x \cos 2 x \sin 3 x=\sin 2 x+\sin 4 x+\sin 6 x$$
Step-by-Step Solution
Verified Answer
Identity verified: both sides simplify to \( \sin 2x + \sin 4x + \sin 6x \).
1Step 1: Express the Left-Hand Side
We start with the expression on the left-hand side, which is given by \( 4 \cos x \cos 2x \sin 3x \). Remember that trigonometric identities can help simplify expressions by rewriting them in different forms.
2Step 2: Apply Product-to-Sum Identity
We use the product-to-sum identities. The identity for a product of sine and cosine is: \[ 2 \cos A \sin B = \sin (A + B) - \sin (A - B) \] Firstly, apply the identity to \( \cos 2x \sin 3x \): \[ 2 \cos 2x \sin 3x = \sin(2x + 3x) - \sin(2x - 3x) \] \[ = \sin 5x - \sin(-x) \] Thus, \( 2 \cos 2x \sin 3x = \sin 5x + \sin x \).
3Step 3: Continue Simplification Using Identities
Next, substitute back into the main expression: \[ 4 \cos x (\sin 5x + \sin x) = 4 \cos x \sin 5x + 4 \cos x \sin x \] Again, apply the product-to-sum identities: \[ 2 \cos x \sin 5x = \sin(6x) - \sin(4x) \] \[ 2 \cos x \sin x = \sin(2x) \]Combine these results: \[ 4 \cos x \sin 5x + 4 \cos x \sin x = 2(\sin 6x - \sin 4x) + 2(\sin 2x) \]\[ = 2\sin 6x - 2\sin 4x + 2\sin 2x \] \[ = \sin 6x + \sin 2x + \sin 4x \].
4Step 4: Conclude with Verification
The expression derived is \( \sin 2x + \sin 4x + \sin 6x \), which matches the right-hand side of the original identity given. This confirms that the identity holds true through proper simplification and application of trigonometric identities.
Key Concepts
PrecalculusTrigonometric SimplificationProduct-to-Sum Identities
Precalculus
Precalculus serves as a bridge between algebra and calculus, ensuring students have the necessary skills to tackle more advanced topics. One fundamental component of precalculus is the study of trigonometric functions. These functions show the relationships between the angles and sides of triangles and are pivotal in understanding periodic phenomena.
In precalculus, the emphasis is placed on mastering trigonometric identities and functions. These may appear complex at first, but they can be enormously rewarding to understand. Mastery in this area aids in simplifying expressions, solving equations, and even modeling real-world situations like sound waves or satellite trajectories. This foundational knowledge is necessary to grasp calculus concepts effectively.
Therefore, it is important to practice simplifying expressions and equations regularly, using these trigonometric identities to build a solid precalculative foundation.
In precalculus, the emphasis is placed on mastering trigonometric identities and functions. These may appear complex at first, but they can be enormously rewarding to understand. Mastery in this area aids in simplifying expressions, solving equations, and even modeling real-world situations like sound waves or satellite trajectories. This foundational knowledge is necessary to grasp calculus concepts effectively.
Therefore, it is important to practice simplifying expressions and equations regularly, using these trigonometric identities to build a solid precalculative foundation.
Trigonometric Simplification
Trigonometric simplification is a powerful tool in precalculus, used to manipulate expressions into simpler forms. By doing so, students can solve trigonometric equations more efficiently or verify identities, as seen in the given exercise.
Simplification involves using formulas and identities to rewrite expressions. For example, in our exercise, the product-to-sum identities were employed to transform a complex product of functions into a more manageable sum. This technique reduces the complexity of expressions and makes them easier to compare or work with.
Key points for successful simplification include recognizing applicable identities, performing logical algebraic operations, and keeping track of positive and negative signs. Each step should be done carefully to ensure the accuracy of the final simplified expression.
Simplification involves using formulas and identities to rewrite expressions. For example, in our exercise, the product-to-sum identities were employed to transform a complex product of functions into a more manageable sum. This technique reduces the complexity of expressions and makes them easier to compare or work with.
Key points for successful simplification include recognizing applicable identities, performing logical algebraic operations, and keeping track of positive and negative signs. Each step should be done carefully to ensure the accuracy of the final simplified expression.
Product-to-Sum Identities
Product-to-sum identities are an essential type of trigonometric identity. They simplify expressions involving products of trigonometric functions by converting them into sums or differences.
These identities include formulas like
Moreover, mastering these types of identities prepares students for calculus and helps in understanding the finer details of trigonometric functions. Using them effectively can save time and effort, making mathematical problem-solving more intuitive and streamlined.
These identities include formulas like
- \( 2 \cos A \sin B = \sin(A + B) - \sin(A - B) \)
- \( 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \)
- \( 2 \sin A \sin B = \cos(A - B) - \cos(A + B)\)
Moreover, mastering these types of identities prepares students for calculus and helps in understanding the finer details of trigonometric functions. Using them effectively can save time and effort, making mathematical problem-solving more intuitive and streamlined.
Other exercises in this chapter
Problem 22
Find all solutions of the equation. $$4 \cos \theta-2=0$$
View solution Problem 23
Write the expression as an algebraic expression in \(x\) for \(x>0\). $$\sin \left(\tan ^{-1} x\right)$$
View solution Problem 23
Verify the Identity. $$\tan ^{4} k-\sec ^{4} k=1-2 \sec ^{2} k$$
View solution Problem 23
Verify the identity. \(\sec 2 \theta=\frac{\sec ^{2} \theta}{2-\sec ^{2} \theta}\)
View solution