Evaluate
arctan(cot10Pi/9)
sec(arccos(-1/6)
csc(arcsec(5/3)
sin(2 arccos(sqrt5/5)
Try solving the problems first on your own and check if the answer you got is correct.
tips: Simplify the form like sec(x)to 1/cos(x). if the problem is the arctan, arcsin, and the likes, try to convert the equation into something like this:
x = arctan (cot(10pi/9))
tan(x) = cot(10pi/9)
then find the arctan of both sides once you simplified the right side.
arctan(cot(10pi/9))= 7pi/18
sec(arccos(-1/6))= -6
csc(arcsec(5/3))= 5/4
sin(2*arccos(sqrt(5)/5))= 4/5
Good luck solving! :D
To evaluate the given expressions, let's break them down step by step:
1. Evaluate arctan(cot(10π/9)):
- First, determine the value of cot(10π/9). Since cot(x) = 1/tan(x), we can rewrite this as cot(10π/9) = 1/tan(10π/9).
- Next, evaluate tan(10π/9). To find the value of tan(x), we can use the unit circle or trigonometric identities.
- In this case, 10π/9 is an obtuse angle in the second quadrant. The reference angle is 7π/9, and we know that tan(x) = sin(x)/cos(x). Therefore, tan(10π/9) = sin(7π/9)/cos(7π/9).
- Use a calculator to find the values of sin(7π/9) and cos(7π/9), then calculate tan(10π/9) accordingly.
- Finally, find arctan(cot(10π/9)) by using the inverse tangent function (arctan). This will give you the angle in radians.
2. Evaluate sec(arccos(-1/6)):
- First, calculate the value of arccos(-1/6) using the inverse cosine function (arccos). This will give you an angle in radians.
- Next, find sec(arccos(-1/6)) by using the reciprocal of the cosine function. Since sec(x) = 1/cos(x), compute 1/cos(arccos(-1/6)) to obtain the value.
3. Evaluate csc(arcsec(5/3)):
- First, determine the value of arcsec(5/3) using the inverse secant function (arcsec). This will give you an angle in radians.
- Next, find csc(arcsec(5/3)) by using the reciprocal of the sine function. Since csc(x) = 1/sin(x), compute 1/sin(arcsec(5/3)) to obtain the value.
4. Evaluate sin(2arccos(sqrt(5/5))):
- Begin by calculating the value of arccos(sqrt(5/5)). Use the inverse cosine function (arccos) to find the angle in radians.
- Next, find 2arccos(sqrt(5/5)) by multiplying the value obtained in the previous step by 2.
- Finally, evaluate sin(2arccos(sqrt(5/5))) using the sine function to find the value.
Please note that it may be necessary to use a calculator to find the trigonometric function values and perform the necessary calculations during the evaluation process.