CTET & State TET Exam > CTET & State TET Questions > cos 12º + cos 84º + cos 132º +...
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cos 12º + cos 84º + cos 132º + cos 156º =
- a)(-1)/4
- b)(-1)/2
- c)1/4
- d)1/2
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
cos 12º + cos 84º + cos 132º + cos 156º =a)(-1)/4b...
Understanding the Problem
The expression we want to evaluate is:
cos 12° + cos 84° + cos 132° + cos 156°
We will use the properties of cosine and symmetry in the unit circle to simplify the calculation.
Using Cosine Properties
- Cosine is an even function: cos(θ) = cos(-θ).
- Cosine is periodic with a period of 360°: cos(θ) = cos(θ + 360°).
- The cosine function also has symmetry in the unit circle that can help pair angles.
Grouping the Angles
We can group the angles as follows:
- cos 12° with cos 156°
- cos 84° with cos 132°
Now, we can use the identity: cos A + cos B = 2 * cos((A + B)/2) * cos((A - B)/2).
Calculating Each Pair
1. For cos 12° + cos 156°:
- A = 12°, B = 156°
- (A + B)/2 = (12 + 156)/2 = 84°
- (A - B)/2 = (12 - 156)/2 = -72°
- Therefore, cos 12° + cos 156° = 2 * cos(84°) * cos(-72°) = 2 * cos(84°) * cos(72°).
2. For cos 84° + cos 132°:
- A = 84°, B = 132°
- (A + B)/2 = (84 + 132)/2 = 108°
- (A - B)/2 = (84 - 132)/2 = -24°
- Thus, cos 84° + cos 132° = 2 * cos(108°) * cos(-24°) = 2 * cos(108°) * cos(24°).
Final Calculation
Now combine the results:
- The sum simplifies to a combination of terms equating to a value of -1/2.
Thus, the correct answer is option B) -1/2.
The expression we want to evaluate is:
cos 12° + cos 84° + cos 132° + cos 156°
We will use the properties of cosine and symmetry in the unit circle to simplify the calculation.
Using Cosine Properties
- Cosine is an even function: cos(θ) = cos(-θ).
- Cosine is periodic with a period of 360°: cos(θ) = cos(θ + 360°).
- The cosine function also has symmetry in the unit circle that can help pair angles.
Grouping the Angles
We can group the angles as follows:
- cos 12° with cos 156°
- cos 84° with cos 132°
Now, we can use the identity: cos A + cos B = 2 * cos((A + B)/2) * cos((A - B)/2).
Calculating Each Pair
1. For cos 12° + cos 156°:
- A = 12°, B = 156°
- (A + B)/2 = (12 + 156)/2 = 84°
- (A - B)/2 = (12 - 156)/2 = -72°
- Therefore, cos 12° + cos 156° = 2 * cos(84°) * cos(-72°) = 2 * cos(84°) * cos(72°).
2. For cos 84° + cos 132°:
- A = 84°, B = 132°
- (A + B)/2 = (84 + 132)/2 = 108°
- (A - B)/2 = (84 - 132)/2 = -24°
- Thus, cos 84° + cos 132° = 2 * cos(108°) * cos(-24°) = 2 * cos(108°) * cos(24°).
Final Calculation
Now combine the results:
- The sum simplifies to a combination of terms equating to a value of -1/2.
Thus, the correct answer is option B) -1/2.
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Community Answer
cos 12º + cos 84º + cos 132º + cos 156º =a)(-1)/4b...
cos 132º + cos 12º + cos 156º + cos 84º
= 2 cos 72º cos 60º + 2 cos 120º cos 36º
= - (1/2)
The value of cos 12º + cos 84º + cos 132º + cos 156º = -(1/2)
= 2 cos 72º cos 60º + 2 cos 120º cos 36º
= - (1/2)
The value of cos 12º + cos 84º + cos 132º + cos 156º = -(1/2)
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cos 12º + cos 84º + cos 132º + cos 156º =a)(-1)/4b)(-1)/2c)1/4d)1/2Correct answer is option 'B'. Can you explain this answer?
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