Problem 9 - Algebra 2

Question

Use an angle sum identity to verify each identity. $$ \sin 2 \theta=2 \sin \theta \cos \theta $$

Step-by-Step Solution

Answer

The calculated solution shows that \(\sin 2 \theta = 2 \sin \theta \cos \theta\). So the given identity is proven to be true.

Step 1: Express \(2 \theta\) as a sum

We start by expressing \(2 \theta\) as \(\theta + \theta\), that simplifies the expression to \(\sin (\theta + \theta)\).

Step 2: Apply the sine of a sum of two angles

In the next step, we apply the sine of the sum of two angles identity to the expression. This results in: \(\sin (\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta\).

Step 3: Simplify the expression

Now, let's simplify the expression. As both terms on the right side of the equation are identical, we get: \(2 \sin \theta \cos \theta\).

Key Concepts

Angle Sum Identities

In trigonometry, angle sum identities are exceptionally useful for simplifying expressions and solving equations involving angles. An angle sum identity provides a formula for calculating trigonometric functions for the sum of two angles. In the case of the sine function, the angle sum identity is expressed as:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
This identity allows us to express the sine of the sum of two angles separately as two individual components. Each component is the product of a sine and a cosine function. By understanding this identity, we can manage more complex trigonometric expressions by breaking them down into simpler parts.

Sine Function

The sine function is one of the primary trigonometric functions, often denoted as \( \sin \theta \). It is defined on the unit circle, where the sine of an angle corresponds to the y-coordinate of the point on the circle. For any angle \(\theta\), its value ranges between -1 and 1.

• The sine function is periodic with a period of \(2\pi\).
• It is an odd function, meaning \( \sin(-\theta) = -\sin \theta \).
Understanding the behavior of the sine function is crucial for tackling trigonometric problems, as it frequently appears in equations and formulas, like the angle sum identity.

Algebraic Manipulation

Algebraic manipulation plays a key role in verifying and simplifying trigonometric identities. After applying an identity, such as the angle sum identity, simplifying the resulting expression often requires combining like terms or factoring.In the given problem, after applying the identity for \( \sin(\theta + \theta) \), we find ourselves with: \( \sin \theta \cos \theta + \cos \theta \sin \theta \). These two terms are like terms and can be combined using simple algebraic manipulation:
\[ \sin \theta \cos \theta + \cos \theta \sin \theta = 2 \sin \theta \cos \theta \]
This simplification showcases the power of algebraic techniques in transforming the sum of terms into a more familiar product form, verifying the original trigonometric identity.