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Chapter 11: Problem 2

The Double-angle Formula for the sine function is \(\sin (2 \theta)=\) ____.

Short Answer

Expert verified
\(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)

Step by step solution

01

Recall the double-angle formula for sine

The double-angle formula for sine is a trigonometric identity that expresses \(\sin(2\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\).
02

Write down the double-angle formula

The double-angle formula for sine is given by \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric identity
Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They help simplify complex trigonometric expressions and solve equations.
The double-angle formula for sine is one such identity.
It is derived from other fundamental trigonometric concepts and provides a compact way to express \(\sin (2\theta) \) in terms of \(\sin (\theta) \) and \(\cos (\theta)\). To fully understand it, reviewing standard identities like Pythagoras鈥� theorem in trigonometry can be useful.
sine function
The sine function, often written as \(\sin \theta\), measures the y-coordinate of a point on the unit circle at an angle \(\theta\) from the positive x-axis.
It is essential in describing waveforms especially in physics and engineering.
For example, the height of a point on a ferris wheel over time can be modeled using a sine function.
In trigonometric identities, \(\sin (\theta)\) frequently appears due to its relationship with the geometry of circles.
cosine function
The cosine function, written as \(\cos \theta\), measures the x-coordinate of a point on the unit circle at an angle \(\theta\) from the positive x-axis.
It is crucial in various scientific and engineering fields.
Similar to the sine function, the cosine function helps model periodic phenomena like sound waves or light intensity.
In the double-angle formula for sine, \(\cos \theta\) plays a vital role by combining with \(\sin \theta\) to express \(\sin (2\theta)\).
double-angle
The double-angle concept refers to trigonometric identities that involve doubling an angle. In the case of the sine function, the double-angle formula is \(\sin (2\theta) = 2\sin (\theta)\cos (\theta)\).
This identity allows us to express trigonometric functions of any double angle using their single-age counterparts.
Understanding this concept is helpful for simplifying expressions and solving complex trigonometric equations.
Using the double-angle formula effectively can simplify a broad range of problems in mathematics and physics.

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