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sin 2000 + cos 2000 is
- a)Zero
- b)Positive
- c)Zero or positive.
- d)Negative
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
sin 2000+ cos 2000 isa)Zerob)Positivec)Zero or positive.d)NegativeCorr...
Because both sin 2000 and cos 2000 lies in 3rd quadrant. In 3rd quadrant values of sin and cos are negative.
This question is part of UPSC exam. View all JEE courses
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sin 2000+ cos 2000 isa)Zerob)Positivec)Zero or positive.d)NegativeCorr...
Understanding the Trigonometric Functions
To solve the expression sin 2000 + cos 2000, we first need to understand the behavior of sine and cosine functions. Both functions oscillate between -1 and 1.
Analyzing sin 2000
- The sine function is periodic with a period of 2π (approximately 6.28).
- To find sin 2000, we can reduce 2000 modulo 2π.
- 2000 radians corresponds to approximately 318.31 full cycles (since 2000 / 2π ≈ 318.31).
- The angle can be simplified as 2000 - 318 * 2π, which gives us a remainder of approximately 5.76 radians.
Analyzing cos 2000
- Similarly, for cos 2000, we apply the same reduction.
- After calculating, we find that cos 2000 also corresponds to the same angle of around 5.76 radians.
Calculating sin(5.76) and cos(5.76)
- At 5.76 radians, which is slightly less than π (approximately 3.14), we are in the third quadrant.
- In the third quadrant, sin is negative, and cos is also negative.
Conclusion
- Hence, both sin(5.76) and cos(5.76) are negative.
- Therefore, sin 2000 + cos 2000 = sin(5.76) + cos(5.76) yields a negative value.
Thus, the correct answer is option 'D' - Negative.
To solve the expression sin 2000 + cos 2000, we first need to understand the behavior of sine and cosine functions. Both functions oscillate between -1 and 1.
Analyzing sin 2000
- The sine function is periodic with a period of 2π (approximately 6.28).
- To find sin 2000, we can reduce 2000 modulo 2π.
- 2000 radians corresponds to approximately 318.31 full cycles (since 2000 / 2π ≈ 318.31).
- The angle can be simplified as 2000 - 318 * 2π, which gives us a remainder of approximately 5.76 radians.
Analyzing cos 2000
- Similarly, for cos 2000, we apply the same reduction.
- After calculating, we find that cos 2000 also corresponds to the same angle of around 5.76 radians.
Calculating sin(5.76) and cos(5.76)
- At 5.76 radians, which is slightly less than π (approximately 3.14), we are in the third quadrant.
- In the third quadrant, sin is negative, and cos is also negative.
Conclusion
- Hence, both sin(5.76) and cos(5.76) are negative.
- Therefore, sin 2000 + cos 2000 = sin(5.76) + cos(5.76) yields a negative value.
Thus, the correct answer is option 'D' - Negative.
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sin 2000+ cos 2000 isa)Zerob)Positivec)Zero or positive.d)NegativeCorrect answer is option 'D'. Can you explain this answer?
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