What is the double angle formula for sin2t?

The double angle formula for sin2t is given by sin2t = 2 sin t cos t.

Understanding the Sine Double Angle Formula

This fundamental identity in trigonometry allows us to express the sine of twice an angle in terms of the sine and cosine of the original angle. Specifically, it states that the sine of a doubled angle is equivalent to twice the product of the cosine and sine of the original angle. This formula is incredibly useful for simplifying complex trigonometric expressions and solving various mathematical problems.

Components of the Formula

Let's break down the elements of the formula sin2t = 2 sin t cos t:

• 2: This represents a scalar multiplier, indicating that the product of sin t and cos t is doubled.
• sin t: This is the sine of the original angle, t. The sine function relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse.
• cos t: This is the cosine of the original angle, t. The cosine function relates an angle of a right-angled triangle to the ratio of the length of the adjacent side to the length of the hypotenuse.

Applications of the Double Angle Formula

The double angle formulas are essential tools in various mathematical contexts, particularly in trigonometry and calculus. Some key applications include:

• Simplifying Expressions: It helps in reducing complex trigonometric expressions to simpler forms, making them easier to manipulate.
• Solving Equations: It's frequently used to solve trigonometric equations where angles are multiples of each other.
• Calculus: In calculus, these formulas are crucial for integrating and differentiating trigonometric functions, especially when dealing with powers of sine or cosine. For instance, to integrate sin^2(t) or cos^2(t), one might use related double angle identities.
• Physics and Engineering: Applied in fields like wave mechanics, signal processing, and oscillations, where understanding periodic phenomena is critical.

Illustrative Example

Let's consider an example to see how the formula sin2t = 2 sin t cos t is applied.

Problem: If you know that sin t = 0.6 and cos t = 0.8, what is the value of sin2t?

Solution:

1. Identify the given values:
   • sin t = 0.6
   • cos t = 0.8
2. Apply the double angle formula:
   • sin2t = 2 sin t cos t
3. Substitute the values:
   • sin2t = 2 * (0.6) * (0.8)
4. Calculate the result:
   • sin2t = 2 * 0.48
   • sin2t = 0.96

This example demonstrates how straightforward it is to calculate the sine of a doubled angle when the sine and cosine of the original angle are known.

For more information on trigonometric identities and their applications, you can refer to comprehensive resources on trigonometry.