Problem 33 - Algebra and Trigonometry

Question

Use the addition formula for sine to prove the double-angle formula for sine.

Step-by-Step Solution

Answer

The double-angle formula \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) is proven by applying the sine addition formula for \( \theta + \theta \).

Step 1: Understand the Formulas Involved

The addition formula for sine is \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). We want to use this formula to prove the double-angle identity for sine: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

Step 2: Set Up the Double Angle

To find \( \sin(2\theta) \), we will apply the addition formula for sine with \( a = \theta \) and \( b = \theta \), thus turning \( a + b \) into \( 2\theta \).

Step 3: Apply the Sine Addition Formula

Using \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \), replace \( a \) and \( b \) with \( \theta \). So we get:\[ \sin(2\theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta) \]

Step 4: Simplify the Expression

Notice that both terms on the right-hand side are the same: \( \sin(\theta)\cos(\theta) \). Therefore, we can combine them:\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]

Step 5: Conclusion

The original expression using the addition formula simplifies correctly to the double angle formula, thus proving that \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) using the addition formula for sine.

Key Concepts

sine addition formula

The sine addition formula is a fundamental identity in trigonometry. It expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles. Specifically, the formula is given by:

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)

This formula is incredibly useful in simplifying complex trigonometric expressions and solving equations. It allows us to break down a function of two angles into simpler parts, which are often easier to manipulate.
Applying the sine addition formula involves substituting the given angles into the formula. By applying it to known values of angles, one can derive other important trigonometric identities, like the double-angle formula for sine, which is very handy in both theoretical work and practical applications, such as in physics or engineering.
In summary, mastering the sine addition formula opens up many possibilities in trigonometry, making it a crucial tool in this field of mathematics.

trigonometric identities

Trigonometric identities are equations that are true for all angles. They involve trigonometric functions and can be used to simplify expressions and solve trigonometric equations. Common trigonometric identities include Pythagorean identities, angle sum and difference identities, and double-angle identities.
The double-angle identity for sine is an example of a trigonometric identity. It states:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

This identity is derived from the sine addition formula by setting the angles \( a \) and \( b \) to be identical, thus simplifying the addition of the angles into a multiplication of sine and cosine for a single angle.
Trigonometric identities are key to proving other mathematical theorems and solving complex trigonometric problems. They reveal deep relationships between trigonometric functions, providing insights into the nature of angles and their corresponding functions, and are useful in various scientific fields.

sine function

The sine function is one of the fundamental trigonometric functions. It is a periodic function that describes the vertical position of a point on the unit circle as it moves around. The function is defined for any real number, making it versatile for various applications.
The sine function is commonly denoted as \( \sin(\theta) \), where \( \theta \) is the angle in radians or degrees. Its values range from -1 to 1, and it has a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.

• The sine function is odd, which implies that \( \sin(-\theta) = -\sin(\theta) \).
• Important values include \( \sin(0) = 0 \), \( \sin(\frac{\pi}{2}) = 1 \), and \( \sin(\pi) = 0 \).

Understanding the sine function is critical for studying oscillations and waves, making it essential in physics, engineering, and many mathematical analyses. Its properties are leveraged in the derivation and application of other trigonometric formulas and identities.