Solving sin(x)cos(x) Given sin(x) - cos(x) 3/4: A Comprehensive Guide

In this article, we will walk through the process of solving for sin(x)cos(x) given the equation sin(x) - cos(x) 3/4. This involves using fundamental trigonometric identities, algebraic manipulation, and the quadratic formula, providing a detailed and comprehensive solution for those interested in understanding such mathematical problems.

Step-by-Step Solution

To solve for sin(x)cos(x) given the equation sin(x) - cos(x) 3/4, let's follow these steps:

Express sin(x) in terms of cos(x)

First, we can express sin(x) in terms of cos(x). From the given equation:

sin(x) cos(x) (3/4)

Use the Pythagorean Identity

Next, we use the Pythagorean identity sin^2(x) cos^2(x) 1 to substitute and solve for cos(x):

(cos(x) (3/4))^2 cos^2(x) 1

Expanding the expression:

cos^2(x) (3/2)cos(x) (9/16) cos^2(x) 1

Combining like terms:

2cos^2(x) (3/2)cos(x) (9/16) - 1 0

This simplifies to:

2cos^2(x) (3/2)cos(x) - (7/16) 0

Multiplying through by 16 to eliminate the fraction, we get:

32cos^2(x) 24cos(x) - 7 0

Use the Quadratic Formula

Using the quadratic formula, where a 32, b 24, and c -7, we find:

cos(x) (-b ± sqrt(b^2 - 4ac)) / (2a)

First, calculate the discriminant:

b^2 - 4ac 24^2 - 4 * 32 * (-7) 576 1472 2048

Then, taking the square root:

sqrt(2048) 32sqrt(2)

Substituting back into the quadratic formula, we get:

cos(x) (-24 ± 32sqrt(2)) / 64

Which simplifies to:

cos(x) (-3 ± 2sqrt(2)) / 8

Find sin(x)

Using sin(x) cos(x) (3/4), we find sin(x) for each value of cos(x):

cos(x) (-3 2sqrt(2)) / 8 implies sin(x) (3 - 2sqrt(2)) / 8 cos(x) (-3 - 2sqrt(2)) / 8 implies sin(x) (3 2sqrt(2)) / 8

Calculate sin(x)cos(x)

Now, to find sin(x)cos(x), we use the values of sin(x) and cos(x):

sin(x)cos(x) ((-3 - 2sqrt(2)) / 8) * ((3 - 2sqrt(2)) / 8) (7 / 32)

Alternative Method: Using Angle Transformation

Alternatively, we can solve this using an angle transformation. Given the equation:

-cos(x) sin(x) 3/4

We can write this as:

sqrt{2} * (cos(x - 135°) 3/4)

This gives us:

cos(x - 135°) (3/4) / sqrt{2}

Let θ 135° - x, then:

2x 270° - 2θ

Thus:

sin(x)cos(x) (1/2)sin(2x) (1/2)sin(270° - 2θ) -1/2cos(2θ)

Using the double angle formula:

cos(2θ) 2cos^2(θ) - 1

Since cos(θ) 3(2) / 8 3sqrt(2) / 8, we get:

2(3sqrt(2)/8)^2 - 1 54/64 - 1 -7/32

Thus, sin(x)cos(x) -1/2 * (54/64 - 1) 7/32

Conclusion

In conclusion, the value of sin(x)cos(x) given the equation sin(x) - cos(x) 3/4 is 7/32. We demonstrated two different methods: one algebraic and another using angle transformations. Both methods confirm the same result, providing a comprehensive understanding of solving such trigonometric equations.

", "meta_description": "Learn how to solve for sin(x)cos(x) given sin(x) - cos(x) 3/4 using trigonometric identities, algebraic manipulation, and the quadratic formula. Explore both algebraic and angle transformation methods for a thorough understanding.