Hi I have a Trig question that i don't understand, can someone please explain how to solve for it using cosine law??
A Clock with a radius of 15 cm has an 11 cm minute hand and a 7 cm hour hand. How far apart, to the nearest centimetre, are the tips of the hands at each time?
a) 3:30 pm b) 6:38 am
Consider a starting time of 12:00.
Every increment of time creates a new triangle with sides of 7 and 11 and an included angle of 5.4m which can be solved using the Law of Cosines.
Where did the 5.4 m come from?
The minute hand of a clock moves 360/60 = 6º/minute or 6º/m.
The hour hand of a clock moves 30/60 = .50º/m.
For example, at 25 minutes after 12:00, the minute hand has rotated 6(25) = 150º while the hour hand has rotated .50(25) = 12.5º. Therefore, the two hands are now 150 - 12.5 = 137.5º apart.
In general, the angle µ between the two hands is 5.5m up to the point when they are 180º apart.
We therefore, now have a triangle with sides of 7 and 11 and an included angle of 137.5º.
Using the Law of Cosines, a^2 = b^2 + c^2 - 2abcosA or a^2 = 7^2 + 11^2 - 2(7)11(cos137.5) = 16.83cm.
What happens beyond the point where the two hands are 180º apart you ask?
When 5.5m is greater than 180º, you subtract the result from 360º to derive the angle between the two hands.
Thus, at 12:40, µ = 360 - 5.5(40) = 140º.
At 1:00, the two hands are 30º apart.
After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.
You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - .5m = 180 or m = 32.7272 minutes after 1:00.
I'll let you continue from here.
A slight correction to the last expression result in my previous response.
After 1:00, the distance between the tips of the hands continues to decrease until they are 4cm apart when the hour hand has rotated to 30 + .5m and the minute hand has rotated to 6m.
At this point 30 + .5m = 6m or m = 5.4545 minutes after 1:00, the point in time when the two hands are coincident.
You can now continues in the same manner as you did earlier from 1:5.4545 to the point where the two hands are again 180º apart when 6m - (30 + .5m) = 180 or m = 38.1818 minutes after 1:00.
I'll let you continue from here.
find in degrees and radians the angle between the hour hand and the minute hand of a clock at half past three
To solve this question using the cosine law, we first need to understand the concept of the cosine law.
The cosine law is a formula used to find the angle or side length of a triangle when we know the lengths of two sides and the included angle, or when we know the lengths of all three sides. The formula is as follows:
c² = a² + b² - 2ab * cos(C)
where c is the side opposite to angle C, and a and b are the lengths of the other two sides of the triangle.
Now let's apply the cosine law to solve the given question.
a) 3:30 pm:
To find the distance between the tips of the clock hands at 3:30 pm, we need to consider the positions of the hour and minute hands. At 3:30 pm, the minute hand will point at the 6 on the clock, which is 180 degrees. The hour hand will be halfway between 3 and 4, which is 7.5 on the clock or 225 degrees.
Let's consider the triangle formed by the center of the clock, the tip of the minute hand, and the tip of the hour hand. We know the lengths of the sides adjacent to the angles we need to find.
Using the cosine law, we can find the distance between the tips of the hands as follows:
d² = (11² + 7²) - 2(11)(7) * cos(225)
Now we can calculate using the cosine law formula:
d² = 121 + 49 - 2 * 11 * 7 * cos(225)
d² ≈ 170.43
To find the actual distance, we take the square root of this result:
d ≈ √170.43 ≈ 13.05 cm
Therefore, the distance between the tips of the clock hands at 3:30 pm is approximately 13.05 cm.
b) 6:38 am:
Similarly, for the time 6:38 am, the minute hand will be at 38 minutes past the hour, which is 228 degrees. The hour hand will be at 6-hours, which is 180 degrees.
Using the same cosine law formula, we can find the distance as follows:
d² = (11² + 7²) - 2(11)(7) * cos(228)
Now we can calculate:
d² = 121 + 49 - 2 * 11 * 7 * cos(228)
d² ≈ 168.63
Taking the square root of this result gives us:
d ≈ √168.63 ≈ 12.98 cm
Therefore, the distance between the tips of the clock hands at 6:38 am is approximately 12.98 cm.