Find the exact value of cot(arcsin(12/13))
and
cos(arcsin(1.7/2))
I know that cos(arcsin(x))=sin(arccos(x))=sqrt(1-x^2). I'm having more difficulty with the first one. Please help! Thank you
I got the second answer. Still confused on the first one.
draw your triangle You have the opposite leg=12 and the hypotenuse=13
So, the adjacent leg is √(169-144) = 5
cot(x) = 5/12
300
To find the exact value of cot(arcsin(12/13)), we need to recall the relationship between the trigonometric functions sine, cosine, and cotangent.
The cotangent of an angle is the reciprocal of the tangent of that angle, which is equal to the ratio of the sine to the cosine of that angle.
For the given expression, cot(arcsin(12/13)), we can start by finding the sine and cosine of arcsin(12/13) and then taking their ratio.
1. Let's find the sine of arcsin(12/13):
We know that arcsin(12/13) is an angle whose sine is 12/13. So we can say that sin(arcsin(12/13)) = 12/13.
2. Now, to find the cosine of arcsin(12/13), we can use the Pythagorean identity:
cos(arcsin(x)) = sqrt(1 - sin^2(arcsin(x))).
Substituting x = 12/13 into the formula, we have:
cos(arcsin(12/13)) = sqrt(1 - (12/13)^2).
Now, we can calculate the value:
cos(arcsin(12/13)) = sqrt(1 - (144/169)).
= sqrt((169 - 144)/169).
= sqrt(25/169).
= 5/13.
3. Finally, we can now find the cotangent by taking the reciprocal of the tangent, which is the ration of sine to cosine:
cot(arcsin(12/13)) = 1 / tan(arcsin(12/13)).
Since tan(arcsin(x)) = sqrt(1 - sin^2(arcsin(x))) / sin(arcsin(x)), we have:
tan(arcsin(12/13)) = sqrt(1 - (12/13)^2) / (12/13).
Plugging in the values, we get:
cot(arcsin(12/13)) = 1 / (sqrt(1 - (12/13)^2) / (12/13)).
= (12/13) / sqrt(1 - (12/13)^2).
Simplifying this expression, we have:
cot(arcsin(12/13)) = 12 / sqrt(1 - (12/13)^2).
So, the exact value of cot(arcsin(12/13)) is 12 / sqrt(1 - (12/13)^2).
Now, let's move on to the second question.
To find the exact value of cos(arcsin(1.7/2)), we will apply the same principles.
1. Let's find the sine of arcsin(1.7/2):
Since arcsin(1.7/2) is an angle whose sine is 1.7/2, sin(arcsin(1.7/2)) = 1.7/2.
2. To determine the cosine of arcsin(1.7/2), we use the Pythagorean identity:
cos(arcsin(x)) = sqrt(1 - sin^2(arcsin(x))).
Substituting x = 1.7/2, we have:
cos(arcsin(1.7/2)) = sqrt(1 - (1.7/2)^2).
Now, we can calculate the value:
cos(arcsin(1.7/2)) = sqrt(1 - (2.89/4)).
= sqrt((4 - 2.89)/4).
= sqrt(1.11/4).
= sqrt(111/400).
= sqrt(111)/sqrt(400).
= sqrt(111)/20.
So, the exact value of cos(arcsin(1.7/2)) is sqrt(111)/20.