Solving the Equation sinθcosθ 1 for θ in [0°, 90°]

The equation sinθcosθ 1 might seem daunting at first glance, but it can be solved by using trigonometric identities and a bit of algebraic manipulation. We will explore how to solve this equation for in the range of [0°, 90°].

Step 1: Substitution Using the Pythagorean Identity

To start, we use the Pythagorean identity which states that sin^2θ cos^2θ 1. We can rearrange the original equation as follows:

sinθcosθ  1

Rearranging gives us:

sinθ  1 - cosθ

We then square both sides to eliminate the sine function:

sin^2θ  (1 - cosθ)^2

Using the identity sin^2θ 1 - cos^2θ, we substitute:

1 - cos^2θ  (1 - cosθ)^2

Simplifying the right-hand side:

1 - cos^2θ  1 - 2cosθ   cos^2θ

Bringing all terms to one side:

2cos^2θ - 2cosθ  0

We can factor out 2cosθ:

2cosθ(cosθ - 1)  0

This gives us two potential solutions:

(cosθ 0), which is not within the range [0°, 90°] (cosθ 1), which corresponds to θ 0°

However, θ 0° is not in the given range [0°, 90°].

Step 2: Finding the Correct Solution

Going back to the original equation, we need to find the value of θ such that both sinθ and cosθ are equal to , which occurs at θ 45°. This is because:

sin45°  cos45°  frac{sqrt{2}}{2}

Hence, the only solution within the range [0°, 90°] is:

θ  45°

Conclusion

The value of θ for which sinθcosθ 1 in the range [0°, 90°] is:

θ  45°

In summary, by using the Pythagorean identity and some algebraic manipulation, we can solve the equation and find the correct value of θ. Understanding these steps can be crucial for more complex problems involving trigonometric equations.