1. John's clock is broken. The minute hand rotates around the clock correctly, but the hour hand is stuck in the three o'clock position. Suppose John first looks at the clock when the hands are
aligned and it shows 3:15. He looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring -135 degrees. How many degrees could the minute
hand have rotated to reach its current position?
-45
-325
-405
-495
2. 5. If sin0 =√2/2 which could not be the value of 0.
225
45
135
405
3. For which value of 0 is cot 0 equal 0
pi
2pi
pi/
0
pi/2
Can you please answer Question 1- 3
1. John's clock is broken. The minute hand rotates around the clock correctly, but the hour hand is stuck in the three o'clock position. Suppose John first looks at the clock when the hands are
aligned and it shows 3:15. He looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring -135 degrees. How many degrees could the minute
hand have rotated to reach its current position?
-45
-325
-405
-495
2. 5. If sin0 =√2/2 which could not be the value of 0.
225
45
135
405
3. For which value of 0 is cot 0 equal 0
pi
2pi
pi/
0
1. The answer is -45 degrees.
To solve this problem, we need to use the fact that the minute hand rotates 360 degrees in 60 minutes, or 6 degrees per minute. Since John first looked at the clock when the hands were aligned at 3:15, we know that the minute hand had rotated 15 minutes * 6 degrees/minute = 90 degrees from the 12 o'clock position.
Now, let x be the number of degrees the minute hand has rotated from the 12 o'clock position to reach its current position. Then, the hour hand has rotated (3 + x/360) * 30 degrees clockwise from the 12 o'clock position (since it is stuck at the 3 o'clock position).
At the second observation, the hour and minute hands form an angle of -135 degrees, or 225 degrees counterclockwise from the 3 o'clock position. Therefore, we have:
(3 + x/360) * 30 - x = 225
90 - x/12 - x = 225
-x/12 = 135
x = -45
So the minute hand has rotated -45 degrees from the 12 o'clock position to reach its current position.
2. The answer is 405.
We know that sin 45 degrees = √2/2, so one possible value of 0 is 45 degrees. However, sin is a periodic function with period 360 degrees, so we can add or subtract any multiple of 360 from 45 degrees to get other values of 0 that satisfy sin 0 = √2/2.
45 + 360k and 315 + 360k (where k is an integer) are all possible values of 0 that satisfy sin 0 = √2/2.
However, 405 is not a possible value of 0 that satisfies this equation.
3. The answer is pi/2.
Recall that cot 0 = cos 0 / sin 0. Therefore, cot 0 = 0 when cos 0 = 0 (since sin 0 cannot be 0).
Cosine is equal to 0 at all odd multiples of pi/2. Therefore, the possible values of 0 that satisfy cot 0 = 0 are:
pi/2, 3pi/2, 5pi/2, ...
The only value of 0 in the interval [0, 2pi) is pi/2.
1. To find the number of degrees the minute hand could have rotated, we need to determine the angle between 3:15 and the current position of the hour hand.
From 3:15 to the current position, the hour hand would have moved 360 degrees (representing a full rotation) plus an additional fraction of 360 degrees proportional to the minutes passed.
Since the time difference is 15 minutes, we can calculate the fraction of 360 degrees as (15/60) * 360 = 90 degrees.
Therefore, the total angle rotated by the hour hand is 360 + 90 = 450 degrees.
The given angle between the hour and minute hands is -135 degrees.
To find the degree the minute hand could have rotated, we calculate the difference between the hour hand's rotation (450 degrees) and the given angle (-135 degrees):
450 - (-135) = 450 + 135 = 585 degrees.
However, since the minute hand's rotation is limited by a 12-hour cycle, we need to find an equivalent angle within that range.
To find the equivalent angle within the 12-hour cycle, we take the remainder of 585 divided by 360:
585 % 360 = 225 degrees.
Therefore, the minute hand could have rotated by 225 degrees from the aligned position to reach its current position.
Answer: 225 degrees.
2. If sin 0 = √2/2, we can determine the possible values of 0 using the unit circle.
The unit circle represents all the values of sine and cosine for different angles.
For sin 0 to be equal to √2/2, the angle 0 should correspond to the points on the unit circle where the y-coordinate is √2/2.
The angles where sin 0 = √2/2 are 45 degrees and 135 degrees.
Therefore, the value of 0 cannot be 225 degrees.
Answer: 225 degrees.
3. For cot 0 to be equal to 0, the angle 0 should have a tangent equal to infinity or undefined.
The tangent function is undefined at angles where the sine is equal to 0 because division by zero is undefined.
Therefore, the value of 0 should be such that sin 0 = 0.
The angles where sin 0 = 0 are multiples of pi, such as 0, pi, 2pi, and so on.
Therefore, the value of 0 that makes cot 0 equal to 0 is pi.
Answer: pi.