Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin 8 t+\sin 2 t $$

Step-by-Step Solution

Answer

\( \sin 8t + \sin 2t = 2 \sin 5t \cos 3t \)

Step 1: Identify Trigonometric Identity

We need to find an identity that allows us to express \( \sin A + \sin B \) as a sum or difference. The sum-to-product identities can be helpful here. The relevant identity is: \[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

Step 2: Apply the Identity with Given Angles

Using the identity found, set \( A = 8t \) and \( B = 2t \).Substitute these values into the identity: \[ \sin 8t + \sin 2t = 2 \sin \left( \frac{8t+2t}{2} \right) \cos \left( \frac{8t-2t}{2} \right) \]

Step 3: Simplify the Expression Inside the Identity

Simplify the expressions for the sine and cosine within the identity:\[ \frac{8t + 2t}{2} = \frac{10t}{2} = 5t \]\[ \frac{8t - 2t}{2} = \frac{6t}{2} = 3t \]

Step 4: Write the Final Expression

Substitute back into the identity to find the expression as a product:\[ \sin 8t + \sin 2t = 2 \sin 5t \cos 3t \]

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are a fascinating part of trigonometry. They allow us to turn sums and differences of sine or cosine functions into products. This simplifies complex expressions and equations. These identities are very useful in various branches of mathematics, including calculus and physics.
For the case of summing two sine functions, like in our exercise, we use the identity:

• If you have \( \sin A + \sin B \), you can express it as a product: \[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

The beauty of this identity is that it transforms the addition of waves (sine functions) into multiplication of simpler functions (sine and cosine). Understanding these identities makes it much easier to manipulate and solve trigonometric expressions.

Sine Function

The sine function, denoted as \( \sin \), is one of the fundamental trigonometric functions. This function takes an angle as input and returns the ratio of the opposite side to the hypotenuse in a right-angled triangle. But the sine function is more than just a ratio. It’s an oscillating wave that appears in countless natural phenomena such as sound waves and light waves.
In our exercise, we encountered \( \sin 8t \) and \( \sin 2t \). By using trigonometric identities, we transformed the sum of these two functions into a more manageable expression. The new expression, \( 2 \sin 5t \cos 3t \), captures the same wave behavior but in a different form. This manipulation can help in further calculations where dealing with products is more convenient than sums.

Mathematics Education

Mathematics education strives to make complex ideas understandable. Trigonometric identities, like the sum-to-product identities, play a crucial role. They show students how different forms of trigonometric functions relate to each other. This revelation can be crucial not only in understanding mathematics but also in solving real-world problems.
By engaging with problems that require these identities, students learn valuable problem-solving skills. It's more than just applying formulas — it’s about seeing the beauty and interconnections in mathematics. Such exercises boost students' confidence and prepare them for advanced studies.

• Understanding the theory behind the equations.
• Applying knowledge to solve practical problems.
• Developing a deeper appreciation for mathematical symmetry and beauty.

By mastering these concepts, students gain a holistic understanding of trigonometry, which is essential in their educational journey.