Question

Write the given expression as a product of two trigonometric functions of different frequencies. \(\sin 7 t-\sin 6 t\)

Short answer

Short Answer: The given expression, \(\sin 7t - \sin 6t\), can be rewritten as the product of two trigonometric functions of different frequencies using the identity \(2\sin(A)\cos(B) = \sin(A + B) + \sin(A - B)\). After solving for A and B, we find that \(A = \frac{13t}{2}\) and \(B = \frac{t}{2}\). Therefore, the expression can be rewritten as: \(\sin 7t - \sin 6t = 2\sin(\frac{13t}{2})\cos(\frac{t}{2})\).

Step-by-step solution

Understand the Problem

We need to transform the expression \(\sin 7t - \sin 6t\) into the form \(2\sin(A)\cos(B) = \sin(A + B) + \sin(A - B)\). We will focus on manipulating the sum and difference of angles to find A and B.

Rewrite the expression as the sum of the products of trigonometric functions

Using the identity, rewrite the expression: $$2\sin(A)\cos(B) = \sin(A + B) + \sin(A - B)$$ We want to find A and B such that: $$\sin 7t - \sin 6t = \sin(A + B) + \sin(A - B)$$

Comparing the given expression with the identity

Now, let's compare this expression to the given one: $$\sin 7t - \sin 6t = \sin(A + B) + \sin(A - B)$$ We have to match the terms, so let's choose: $$A + B = 7t$$ $$A - B = 6t$$

Solve for A and B values

Using the system of equations, we can solve for A and B: $$A + B = 7t$$ $$A - B = 6t$$ Add the two equations: $$2A = 13t$$ $$A = \frac{13t}{2}$$ Next, subtract the second equation from the first: $$2B = t$$ $$B = \frac{t}{2}$$ Thus, A and B have been found: $$A = \frac{13t}{2}$$ $$B = \frac{t}{2}$$

Rewrite the given expression in the desired form

Now, we can rewrite the given expression in terms of A and B: $$\sin 7t - \sin 6t = \sin(A + B) + \sin(A - B) = 2\sin(A)\cos(B)$$ Using the values of A and B we found in Step 4, this becomes: $$\sin 7t - \sin 6t = 2\sin(\frac{13t}{2})\cos(\frac{t}{2})$$ So, the given expression can be written as a product of two trigonometric functions of different frequencies: $$\sin 7t - \sin 6t = 2\sin(\frac{13t}{2})\cos(\frac{t}{2})$$

Key concepts

Sum to Product Formulas

Sum to product formulas are trigonometric identities that transform the sum or difference of two trigonometric functions into a product of two other trigonometric functions. They are particularly useful when simplifying complex trigonometric expressions or when solving trigonometric equations. The sum to product formulas for sine and cosine are given by:

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

In the exercise provided, we applied the formula for the difference of two sine functions to express \(\sin 7t - \sin 6t\) as a product. This approach is beneficial for calculating the expressions more efficiently and is a key skill in advanced trigonometry.

Trigonometric Functions

Trigonometric functions are foundational to the study of geometry and periodic phenomena. They relate the angles of a triangle to the lengths of its sides and extend this relationship to circular functions that define values for all real numbers. The six fundamental trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Each of these functions has a specific domain and range, and they possess unique properties and graphs. They are periodic, meaning they repeat their values in regular intervals, which makes them particularly important for modeling repetitive occurrences such as sound waves, light waves, and the movement of pendulums.

Understanding these functions is critical not only for solving basic trigonometric equations but also for delving into more complex applications such as Fourier analysis and solving differential equations.

Angle Addition and Subtraction

The angle addition and subtraction formulas are vital tools in trigonometry. These formulas allow us to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. The angle addition formulas for sine and cosine are given by:

• \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
• \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)

These formulas are essential for simplifying trigonometric expressions and solving problems without directly knowing the sine and cosine of the sum or difference of angles. In our exercise, the conversion of \(\sin 7t - \sin 6t\) into a product involved using these concepts to find suitable angles A and B that satisfy the sum to product identities.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy an equation involving trigonometric functions. This process can involve a range of techniques, including graphing the functions, using identities like the Pythagorean identity, double-angle formulas, half-angle formulas, and sum to product formulas.

In the exercise, we encountered a trigonometric equation \(2\sin(A)\cos(B) = \sin(7t) - \sin(6t)\).By employing the sum to product formula, we successfully transformed a trigonometric equation into a simpler format that can be evaluated for specific values of the variable. These strategies form part of a larger set of methods for solving various trigonometric equations, which can feature quadratic forms, multiple angles, or product-to-sum forms.

After identifying the general solution, it is important to consider the domain of the original problem to find the relevant solutions within the specified range. The familiarity with periodic properties and symmetries of trigonometric functions greatly facilitates this task.