Solving the Equation: x tan 45° cos 60° sin 60° cot 60°
Welcome to our detailed guide on solving the trigonometric equation x tan 45° cos 60° sin 60° cot 60°. This article provides a step-by-step approach and explanation of the trigonometric identities involved. Whether you're a student, a teacher, or anyone interested in mathematics, this guide will clarify the process and help you understand the underlying concepts.
Understanding Trigonometric Functions
Before tackling the equation directly, it's essential to refresh our understanding of the basic trigonometric functions involved here:
tan 45° - The tangent of 45 degrees is 1. cos 60° - The cosine of 60 degrees is 1/2. sin 60° - The sine of 60 degrees is √3/2. cot 60° - The cotangent of 60 degrees is 1/√3.Solving the Equation Step-by-Step
Let's substitute these values into the equation x tan 45° cos 60° sin 60° cot 60° and solve for x.
Substituting the Values
Substitute the known values into the equation:
x · tan 45° · cos 60° sin 60° · cot 60°
This simplifies to:
x · 1 · 1/2 √3/2 · 1/√3
Simplifying the Right Side of the Equation
Simplify the right-hand side:
√3/2 · 1/√3 1/2
This is because the √3 terms cancel out:
√3/2 · 1/√3 1/2
Solving for x
Now, the equation simplifies to:
x · 1/2 1/2
Multiply both sides by 2 to isolate x:
x 1
Conclusion and Verification
The value of x is 1. To verify, you can substitute x 1 back into the original equation:
1 · tan 45° · cos 60° sin 60° · cot 60°
This results in:
1 · 1 · 1/2 √3/2 · 1/√3
Which simplifies to:
1/2 1/2
Both sides are equal, confirming that x 1 is the correct solution.
Additional Resources and Tips
If you want to delve deeper into trigonometry, here are some additional resources:
MathIsFun Trigonometry Khan Academy Trigonometry Mathplanet Trigonometric Functions and EvaluatingRemember, trigonometric functions are fundamental in many fields, including physics, engineering, and architecture. Understanding these basics can open up a vast range of applications and problem-solving skills.