Trigonometric Values and the Cosine Addition Formula in Trigonometry
Understanding the relationships between trigonometric functions is crucial for solving a wide range of problems in mathematics and physics. This article focuses on how to find the value of #954;(cos , alpha , beta) when given specific values of (tan , alpha) and (sin , beta).
Background and Problem Statement
In this context, we are given the values:
(tan , alpha sqrt{3}) (sin , beta frac{1}{2})Using these values, our objective is to determine the value of (cos , alpha , beta).
Step-by-Step Solution
The cosine addition formula for two angles ( alpha ) and ( beta ) is: [cos , (alpha beta) cos , alpha , cos , beta - sin , alpha , sin , beta]
Step 1: Finding (cos , alpha ) and (sin , alpha)
Given (tan , alpha sqrt{3}), we use the identity: [tan , alpha frac{sin , alpha}{cos , alpha} sqrt{3}]
This implies that: [sin , alpha sqrt{3} cos , alpha]
Using the Pythagorean identity: [sin^2 , alpha cos^2 , alpha 1]
Substituting (sin , alpha): [(sqrt{3} cos , alpha)^2 cos^2 , alpha 1]
[3 cos^2 , alpha cos^2 , alpha 1]
[4 cos^2 , alpha 1]
[cos^2 , alpha frac{1}{4}]
[cos , alpha frac{1}{2} , text{(since } alpha text{ is in the first quadrant)}]
Now substituting back to find (sin , alpha):
[sin , alpha sqrt{3} cdot frac{1}{2} frac{sqrt{3}}{2}]
Step 2: Finding (cos , beta ) and (sin , beta)
Given (sin , beta frac{1}{2}), we can find (cos , beta) using the same Pythagorean identity:
[sin^2 , beta cos^2 , beta 1]
[left(frac{1}{2}right)^2 cos^2 , beta 1]
[frac{1}{4} cos^2 , beta 1]
[cos^2 , beta 1 - frac{1}{4} frac{3}{4}]
[cos , beta frac{sqrt{3}}{2} , text{(since } beta text{ is in the first quadrant)}]
Step 3: Calculating (cos , (alpha beta))
Now, we substitute [sin , alpha frac{sqrt{3}}{2}, , cos , alpha frac{1}{2}, , sin , beta frac{1}{2}, , cos , beta frac{sqrt{3}}{2}], into the cosine addition formula:
[cos , (alpha beta) cos , alpha , cos , beta - sin , alpha , sin , beta]
[ left(frac{1}{2}right) left(frac{sqrt{3}}{2}right) - left(frac{sqrt{3}}{2}right) left(frac{1}{2}right)]
[ frac{sqrt{3}}{4} - frac{sqrt{3}}{4} 0]
Final Result:
The value of (cos , (alpha beta)) is 0.
Related Formulas and Considerations
While solving this problem, we use several fundamental trigonometric identities and formulas, such as:
(cos , theta sqrt{1 - sin , theta^2}) (sec , theta sqrt{1 tan , theta^2}) (cos , theta frac{1}{sec , theta}) (sin , theta sqrt{1 - cos , theta^2}) (cos , (alpha beta) cos , alpha , cos , beta - sin , alpha , sin , beta)Additionally, we consider the quadrants in which the angles lie to determine the correct signs of the trigonometric functions. This is crucial as sine and cosine functions can be positive or negative depending on the quadrant.
Conclusion
The value of (cos , alpha , beta) given (tan , alpha sqrt{3}) and (sin , beta frac{1}{2}) is 0. This result is derived by accurately finding the values of (sin , alpha) and (cos , alpha), and (sin , beta) and (cos , beta) using trigonometric identities and considering the quadrant in which the angles lie.