cos θ = 0.7677
(a) the nearest 0.01°
(b) the nearest 1'
these are calculator operations
To find the nearest 0.01° and 1', we can use the inverse cosine function.
(a) To find the nearest 0.01°, we round the value of θ to the nearest 0.01.
cos θ = 0.7677
θ = arccos(0.7677)
θ ≈ 39.76°
Rounding to the nearest 0.01°, θ ≈ 39.76°.
(b) To find the nearest 1', we need to convert degrees to minutes (1° = 60') and round to the nearest minute.
θ ≈ 39.76°
39.76° * 60' = 2385.6'
Rounding 2385.6' to the nearest minute, θ ≈ 2386'.
To find the nearest values in degrees and minutes, you need to convert the given cosine value back into an angle measurement.
(a) To find the nearest 0.01°:
To determine the angle in degrees, you can use the inverse cosine function (arccos) on the given value of 0.7677. In mathematical notation, this is expressed as:
θ = arccos(0.7677)
To get the answer, you can use a calculator or a mathematical software that has the arccosine function. By plugging in 0.7677 into the arccos function, you will get the angle in radians. To convert this into degrees, you multiply the value by 180/π (approximately 57.2958). Therefore,
θ ≈ arccos(0.7677) × (180/π)
The result will be in degrees. Round this result to the nearest 0.01° to find the final value.
(b) To find the nearest 1' (1 arcminute):
To convert degrees into arcminutes, you need to multiply the decimal portion of the degree value by 60. Take the result you obtained from part (a) and multiply it by 60 to get the arcminutes. Round this value to the nearest whole number to find the final value.
Therefore, to summarize:
(a) The nearest 0.01°: Calculate θ using arccos(0.7677) and round the result to the nearest 0.01°.
(b) The nearest 1': Multiply the decimal portion of the degree value obtained in (a) by 60 and round the result to the nearest whole number.