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Answer As an experienced tutor registered on UrbanPro, I'd be glad to help with this math question. UrbanPro is indeed a fantastic platform for finding online coaching and tuition services.
Let's tackle this problem step by step. We're given two conditions:
To find the most general value of θθ satisfying both equations, we need to consider the trigonometric relationships between tangent and cosine functions.
First, let's address the equation tanθ=−1tanθ=−1. This tells us that θθ corresponds to an angle where the tangent is negative and has a magnitude of 1. This occurs in the second and fourth quadrants of the unit circle.
Secondly, cosθ=12cosθ=2
1. This corresponds to an angle where the cosine is positive and equals 122
1, which happens in the first quadrant.
Putting these together, we find that the only quadrant where both conditions are met is the fourth quadrant. This is because in the fourth quadrant, tangent is negative and cosine is positive.
In the fourth quadrant, an angle θθ that satisfies both conditions is given by:
θ=3π4+2πnθ=43π+2πn
where nn is an integer.
So, the most general value of θθ satisfying both equations is 3π443π plus any integer multiple of 2π2π.
This solution provides a comprehensive understanding of the problem and showcases how UrbanPro can connect students with knowledgeable tutors who can guide them through challenging concepts. If you need further clarification or assistance, feel free to ask!
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