Approximate the acute angle θ to the following cos θ = 0.7507
(a) the nearest 0.01°
(b) the nearest 1'
well, θ = 41.3489°
so how do you round that?
To approximate the acute angle θ based on the given value of cos θ = 0.7507, we can make use of the inverse cosine function (also known as arccosine) to find the angle.
(a) Nearest 0.01°:
To find the angle to the nearest 0.01°, we can use the arccosine function in a calculator or a mathematical software.
θ = arccos(0.7507)
θ ≈ 41.36°
So, the approximate angle θ to the nearest 0.01° is 41.36°.
(b) Nearest 1':
To find the angle to the nearest 1', we need to convert the angle value from degrees to minutes (1° = 60').
θ = 41.36°
θ ≈ 41° × 60' = 41° × 60' / 1° = 2460'
So, the approximate angle θ to the nearest 1' is 2460'.
To approximate the acute angle θ from the cosine value, you can use the inverse cosine function (also known as the arccosine function). However, keep in mind that the inverse cosine function returns values between 0° and 180°.
(a) To find the nearest 0.01° approximation, you can follow these steps:
1. Use the inverse cosine function (arccos) to find the angle in radians: θ = arccos(0.7507).
2. Convert the angle from radians to degrees by multiplying it by (180/π): θ (in degrees) = arccos(0.7507) × (180/π).
3. Round the angle to the nearest 0.01°.
For example:
θ = arccos(0.7507) × (180/π)
θ ≈ 41.4091°
Rounded to the nearest 0.01°, θ ≈ 41.41°.
(b) To find the nearest 1' (1 minute of arc) approximation, you can follow these steps:
1. Find the angle θ in degrees by applying the inverse cosine function: θ = arccos(0.7507) × (180/π).
2. Use the modulus operator to obtain the remainder of the angle divided by 1°: remainder = θ % 1.
3. Subtract the remainder from the angle: θ = θ - remainder.
4. Round the angle to the nearest whole number (1°).
5. Add the rounded remainder (converted to minutes) to the angle.
For example:
θ = arccos(0.7507) × (180/π)
θ ≈ 41.4091°
remainder = 41.4091° % 1 = 0.4091°
θ (rounded to the nearest whole number) = 41°
θ ≈ 41° + (0.4091° × 60) ≈ 41° + 24.546' ≈ 41° 24.55'.
Thus, the approximate acute angle θ is:
(a) θ ≈ 41.41° (rounded to the nearest 0.01°).
(b) θ ≈ 41° 24.55' (rounded to the nearest 1 minute).