Problem 74
Question
Prove the identity. $$\sin 8 x=2 \sin 4 x \cos 4 x$$
Step-by-Step Solution
Verified Answer
The identity is proved using the double angle formula for sine.
1Step 1: Recall Double Angle Formula
The double angle formula for sine is \( \sin(2\theta) = 2\sin\theta\cos\theta \). This formula will be useful for proving the given identity as 8x and 4x are multiples of each other.
2Step 2: Apply the Double Angle Formula
We start with the left-hand side of the identity, \( \sin(8x) \). We can express this as \( \sin(2 \cdot 4x) \).Using the double angle formula, we substitute and get:\[sin(8x) = \sin(2 \cdot 4x) = 2 \sin(4x) \cos(4x) \]
3Step 3: Observe that Identity is Proved
We observe that when using the double angle formula on \( \sin(8x) \), it directly translates to the right-hand side of the original identity: \( 2\sin(4x)\cos(4x) \). Thus, the identity is verified successfully.
Key Concepts
Double Angle FormulaSine FunctionCosine Function
Double Angle Formula
The double angle formula is a handy tool in trigonometry that helps us deal with problems involving angles that are multiples of smaller angles. Specifically, the double angle formula for sine is expressed as \( \sin(2\theta) = 2\sin\theta\cos\theta \). This formula is derived from the angle addition formulas and serves an important role in simplifying expressions that involve sine and cosine functions when angles are multiplied.
In essence, the formula allows us to rewrite the sine of a double angle in terms of the product of the sine and cosine of the original angle. Such formulas are incredibly useful for proving identities or simplifying trigonometric equations. By recognizing patterns and applying these formulas, we can often simplify complex expressions with ease. In the given exercise, we've used the double angle formula to prove that \( \sin(8x) = 2\sin(4x)\cos(4x) \), by expressing \( \sin(8x) \) as \( \sin(2 \times 4x) \).
In essence, the formula allows us to rewrite the sine of a double angle in terms of the product of the sine and cosine of the original angle. Such formulas are incredibly useful for proving identities or simplifying trigonometric equations. By recognizing patterns and applying these formulas, we can often simplify complex expressions with ease. In the given exercise, we've used the double angle formula to prove that \( \sin(8x) = 2\sin(4x)\cos(4x) \), by expressing \( \sin(8x) \) as \( \sin(2 \times 4x) \).
Sine Function
The sine function is a fundamental trigonometric function that arises in many areas of mathematics and applied sciences. It depicts the relationship between the angle in a right triangle and the ratio of the opposite side to the hypotenuse. The function is periodic, meaning it repeats its values in regular intervals, specifically every \( 360^\circ \) or \( 2\pi \) radians.
A key property of the sine function is its wave-like shape, characterized by its amplitude and period. In the unit circle, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle. This geometric interpretation is fundamental in understanding how we can use trigonometric identities to manipulate and simplify expressions that include the sine function. In our exercise, knowing how the sine function behaves across different angles helped us verify the identity involving the double angle: \( \sin(8x) = 2\sin(4x)\cos(4x) \).
A key property of the sine function is its wave-like shape, characterized by its amplitude and period. In the unit circle, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle. This geometric interpretation is fundamental in understanding how we can use trigonometric identities to manipulate and simplify expressions that include the sine function. In our exercise, knowing how the sine function behaves across different angles helped us verify the identity involving the double angle: \( \sin(8x) = 2\sin(4x)\cos(4x) \).
Cosine Function
The cosine function is another essential trigonometric function, closely related to the sine function. Like the sine function, it describes the ratio between the side adjacent to an angle in a right triangle and the triangle's hypotenuse. Cosine, too, is periodic, with values repeating every \( 360^\circ \) or \( 2\pi \) radians.
Cosine's graph, a cosine wave, is similar to the sine wave but phase-shifted. Therefore, at any angle, the cosine function provides us with the x-coordinate on the unit circle. This means that the cosine function is useful for many practical applications, including solving trigonometric equations and proving identities such as in the exercise provided.
In our specific scenario, we paired the cosine and sine functions using the double angle formula to demonstrate that \( \sin(8x) \) can be expressed as \( 2\sin(4x)\cos(4x) \). Recognizing how these two functions interrelate allows us to manipulate and confirm complex trigonometric identities seamlessly.
Cosine's graph, a cosine wave, is similar to the sine wave but phase-shifted. Therefore, at any angle, the cosine function provides us with the x-coordinate on the unit circle. This means that the cosine function is useful for many practical applications, including solving trigonometric equations and proving identities such as in the exercise provided.
In our specific scenario, we paired the cosine and sine functions using the double angle formula to demonstrate that \( \sin(8x) \) can be expressed as \( 2\sin(4x)\cos(4x) \). Recognizing how these two functions interrelate allows us to manipulate and confirm complex trigonometric identities seamlessly.
Other exercises in this chapter
Problem 73
Prove the identity. $$\cos ^{2} 5 x-\sin ^{2} 5 x=\cos 10 x$$
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Verify the identity. $$\sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x$$
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Verify the identity. $$\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta$$
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Prove the identity. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$
View solution