Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin 6 \theta+\sin 2 \theta $$

Step-by-Step Solution

Answer

The expression is \( 2 \sin (4\theta) \cos (2\theta) \).

Step 1: Apply Sum-to-Product Identities

To express the expression as a sum or difference, we use the sum-to-product identity: \[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]Here, let \( A = 6\theta \) and \( B = 2\theta \). Substitute these values into the identity.

Step 2: Simplify the Expression

Substitute \( A = 6\theta \) and \( B = 2\theta \) into the identity:\[\sin 6\theta + \sin 2\theta = 2 \sin \left( \frac{6\theta + 2\theta}{2} \right) \cos \left( \frac{6\theta - 2\theta}{2} \right)\]Simplify the fractions:\[\sin 6\theta + \sin 2\theta = 2 \sin (4\theta) \cos (2\theta)\]

Step 3: Write Final Answer

After simplifying, the original expression \( \sin 6\theta + \sin 2 \theta \) can be expressed as:\[2 \sin (4\theta) \cos (2\theta)\] This is the expression as a sum of trigonometric functions.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that relate different trigonometric functions to one another. These identities serve as essential tools in solving trigonometric equations and simplifying expressions. One of the key identities used in trigonometry is the sum-to-product identity. This identity acts as a bridge to transform the sum or difference of two sine functions into a product of sine and cosine functions. The specific sum-to-product identity we used in the exercise is:

• \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \)

This identity helps simplify complex trigonometric expressions and is particularly useful when dealing with sums or differences of angles, as it transforms them into more manageable calculations.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are periodic functions that model the relationships between the angles and sides of right triangles. In our exercise, we focus on the sine function, which is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. These functions are pivotal in representing oscillating phenomena such as sound waves or circular motion.When expressing \( \sin 6\theta + \sin 2\theta \) using sum-to-product identities, we leverage the properties of trigonometric functions. By transforming the sum of sines into a product of sine and cosine, we can simplify and solve trigonometric expressions more efficiently. The trigonometric functions, when used in conjunction, allow us to express complex periodic behaviors in simpler, more intuitive forms.

Algebraic Simplification

Algebraic simplification involves reducing expressions to simpler forms to make them easier to work with. When we applied the sum-to-product identity to the expression \( \sin 6\theta + \sin 2\theta \), we aimed to rewrite and simplify the trigonometric expression. By substituting the given values into the identity, we solved the expression into:

• \( 2 \sin (4\theta) \cos (2\theta) \)

This process involved simplifying the fractions and ensuring that all calculations adhered to algebraic rules. Algebraic simplification is crucial in both mathematics and its applications as it allows us to tackle more challenging problems by transforming them into simpler, more manageable forms. This skill is essential, ensuring clarity and efficiency while solving mathematical equations and expressions.