Hey guys! Let's dive into a trigonometric problem where we need to find the value of sin(θ)cos(θ) given that sin(θ) + cos(θ) = 7/13. This is a classic problem that combines algebraic manipulation with trigonometric identities. Understanding how to solve this kind of problem can be super useful in various fields like physics, engineering, and even computer graphics! So, let's get started and break it down step by step.

Understanding the Problem

At its heart, this problem requires us to relate the given sum of sine and cosine to their product. We're given that sin(θ) + cos(θ) = 7/13, and we need to find the value of sin(θ)cos(θ). The key here is to use a well-known algebraic identity that connects the sum and the product of two terms. By squaring the given equation, we can introduce the product term and use the Pythagorean identity to simplify the expression. This approach will allow us to isolate and solve for the desired product.

Step-by-Step Solution

1. Square the given equation:

   We start with the equation sin(θ) + cos(θ) = 7/13. Squaring both sides of the equation gives us:

   (sin(θ) + cos(θ))^2 = (7/13)^2

   Expanding the left side, we get:

   sin^2(θ) + 2sin(θ)cos(θ) + cos^2(θ) = 49/169

2. Apply the Pythagorean identity:

   Recall the fundamental Pythagorean identity: sin^2(θ) + cos^2(θ) = 1. We can substitute this into our equation:

   1 + 2sin(θ)cos(θ) = 49/169

3. Isolate the product term:

   Now, we want to isolate the term 2sin(θ)cos(θ). Subtract 1 from both sides:

   2sin(θ)cos(θ) = 49/169 - 1

   To combine the terms on the right side, we need to express 1 as a fraction with the same denominator:

   2sin(θ)cos(θ) = 49/169 - 169/169

   2sin(θ)cos(θ) = (49 - 169)/169

   2sin(θ)cos(θ) = -120/169

4. Solve for sin(θ)cos(θ):

   Finally, to find sin(θ)cos(θ), divide both sides by 2:

   sin(θ)cos(θ) = (-120/169) / 2

   sin(θ)cos(θ) = -120 / (169 * 2)

   sin(θ)cos(θ) = -60/169

So, the value of sin(θ)cos(θ) is -60/169.

Alternative Method: Using Trigonometric Identities Directly

Another way to approach this problem is by using trigonometric identities to express sin(θ)cos(θ) directly in terms of sin(θ) + cos(θ). This method can be a bit more advanced but provides a deeper understanding of trigonometric relationships.

1. Recall the identity for (sin(θ) + cos(θ))^2:

   We know that (sin(θ) + cos(θ))^2 = sin^2(θ) + 2sin(θ)cos(θ) + cos^2(θ).

2. Use the Pythagorean identity:

   As before, sin^2(θ) + cos^2(θ) = 1, so we can rewrite the equation as:

   (sin(θ) + cos(θ))^2 = 1 + 2sin(θ)cos(θ)

3. Solve for sin(θ)cos(θ):

   We want to isolate sin(θ)cos(θ), so we rearrange the equation:

   2sin(θ)cos(θ) = (sin(θ) + cos(θ))^2 - 1

   Divide by 2 to solve for sin(θ)cos(θ):

   sin(θ)cos(θ) = [(sin(θ) + cos(θ))^2 - 1] / 2

4. Substitute the given value:

   We are given that sin(θ) + cos(θ) = 7/13. Substitute this value into the equation:

   sin(θ)cos(θ) = [(7/13)^2 - 1] / 2

   sin(θ)cos(θ) = [(49/169) - 1] / 2

   sin(θ)cos(θ) = [(49/169) - (169/169)] / 2

   sin(θ)cos(θ) = [-120/169] / 2

   sin(θ)cos(θ) = -60/169

Again, we find that sin(θ)cos(θ) = -60/169.

Common Mistakes to Avoid

• Forgetting the Pythagorean Identity: One of the most common mistakes is forgetting that sin^2(θ) + cos^2(θ) = 1. This identity is crucial for simplifying the equation and solving for the desired value.
• Incorrectly Squaring the Equation: Make sure to correctly expand (sin(θ) + cos(θ))^2. Remember that (a + b)^2 = a^2 + 2ab + b^2.
• Arithmetic Errors: Be careful with your arithmetic, especially when dealing with fractions. A small mistake can lead to an incorrect answer.
• Not Isolating the Desired Term Correctly: Ensure you isolate sin(θ)cos(θ) properly. Double-check each step to avoid any algebraic errors.

Real-World Applications

Understanding trigonometric relationships like this isn't just about solving math problems; it has practical applications in various fields:

• Physics: In physics, these relationships are used in mechanics to analyze projectile motion, oscillations, and wave phenomena.
• Engineering: Engineers use trigonometric functions to design structures, analyze forces, and work with electrical circuits.
• Computer Graphics: Trigonometry is fundamental in computer graphics for creating realistic 3D models, animations, and simulations.
• Navigation: Navigation systems rely on trigonometric functions to calculate distances, angles, and positions.

Practice Problems

To solidify your understanding, try solving these similar problems:

1. If sin(θ) + cos(θ) = 5/13, find the value of sin(θ)cos(θ).
2. Given sin(θ) + cos(θ) = 1/2, determine the value of sin(θ)cos(θ).
3. If sin(θ) + cos(θ) = √2/2, what is the value of sin(θ)cos(θ)?

By practicing these problems, you'll become more comfortable with manipulating trigonometric identities and solving for different variables.

Conclusion

Finding the value of sin(θ)cos(θ) when given sin(θ) + cos(θ) involves using algebraic manipulation and trigonometric identities. By squaring the given equation, applying the Pythagorean identity, and isolating the desired term, we can solve for sin(θ)cos(θ). Whether you're a student tackling homework or someone interested in the practical applications of trigonometry, mastering these techniques will definitely come in handy. Keep practicing, and you'll become a pro at these types of problems in no time! Remember, the value of sin(θ)cos(θ) when sin(θ) + cos(θ) = 7/13 is -60/169. Keep up the great work!