1) Determine the approximate measure
of all angles that satisfy the following.
Give answers to two decimal places. Use
diagrams to show the possible answers.
a) cos θ = 0.42 in the domain
-π ≤ θ ≤ π
b) tan θ = -4.87 in the domain
- _
π
2
≤ θ ≤ π
c) csc θ = 4.87 in the domain
-360° ≤ θ < 180°
d) cot θ = 1.5 in the domain
-180° ≤ θ < 360°
help with 11c
a) To determine the approximate measure of angles that satisfy cos θ = 0.42 in the domain -π ≤ θ ≤ π, we can use the inverse cosine function (also known as arccosine or cos^(-1)).
Using a calculator, find the inverse cosine of 0.42, which is approximately 1.115 (rounded to three decimal places).
Since the domain is -π ≤ θ ≤ π, the possible angles can be found by adding or subtracting 2π (or 360°) multiples of the found value.
Therefore, the approximate measures of the angles that satisfy cos θ = 0.42 in the given domain are:
θ ≈ 1.115, -1.115, 5.257, -5.257, 7.399, -7.399, ...
b) To find the approximate measure of angles that satisfy tan θ = -4.87 in the domain -π/2 ≤ θ ≤ π, we can use the inverse tangent function (also known as arctangent or tan^(-1)).
Using a calculator, find the inverse tangent of -4.87, which is approximately -1.346 (rounded to three decimal places).
Since the domain is -π/2 ≤ θ ≤ π, the possible angles can be found by adding or subtracting π (or 180°) multiples of the found value.
Therefore, the approximate measures of the angles that satisfy tan θ = -4.87 in the given domain are:
θ ≈ -1.346, -1.496, 1.795, -2.943, 3.142, -4.437, ...
c) To determine the approximate measure of angles that satisfy csc θ = 4.87 in the domain -360° ≤ θ < 180°, we can use the inverse cosecant function (also known as arccosecant or csc^(-1)).
Using a calculator, find the inverse cosecant of 4.87, which is approximately 0.205 (rounded to three decimal places).
Since the domain is -360° ≤ θ < 180°, the possible angles can be found by adding or subtracting 360° multiples of the found value.
Therefore, the approximate measures of angles that satisfy csc θ = 4.87 in the given domain are:
θ ≈ 0.205, 360.205, -359.795, 720.205, -719.795, ...
d) To find the approximate measure of angles that satisfy cot θ = 1.5 in the domain -180° ≤ θ < 360°, we can use the inverse cotangent function (also known as arccotangent or cot^(-1)).
Using a calculator, find the inverse cotangent of 1.5, which is approximately 33.69 (rounded to two decimal places).
Since the domain is -180° ≤ θ < 360°, the possible angles can be found by adding or subtracting 180° multiples of the found value.
Therefore, the approximate measures of the angles that satisfy cot θ = 1.5 in the given domain are:
θ ≈ 33.69, 213.69, -146.31, 393.69, -566.31, ...
To determine the approximate measure of all angles that satisfy the given trigonometric equations, we can use the inverse trigonometric functions. Here's how to find the answers to each part:
a) cos θ = 0.42 in the domain -π ≤ θ ≤ π
To find the angle θ that satisfies this equation, we can use the inverse cosine function (also denoted as arccos or cos^-1). Take the arccos of 0.42 to find the angle.
θ = arccos(0.42)
You can use a scientific calculator or math software to find the arccos of 0.42. The solution will be in radians. Convert it to degrees if necessary by multiplying by 180/π.
b) tan θ = -4.87 in the domain - π/2 ≤ θ ≤ π
To find the angle θ that satisfies this equation, we can use the inverse tangent function (also denoted as arctan or tan^-1). Take the arctan of -4.87 to find the angle.
θ = arctan(-4.87)
Again, use a scientific calculator or math software to find the arctan of -4.87. The solution will be in radians. Convert it to degrees if necessary.
c) csc θ = 4.87 in the domain -360° ≤ θ < 180°
To find the angle θ that satisfies this equation, we use the inverse cosecant function (also denoted as arccsc or csc^-1). Take the arccsc of 4.87 to find the angle.
θ = arccsc(4.87)
Use a scientific calculator or math software to find the arccsc of 4.87. The solution will be in radians. Convert it to degrees if necessary.
d) cot θ = 1.5 in the domain -180° ≤ θ < 360°
To find the angle θ that satisfies this equation, we use the inverse cotangent function (also denoted as arccot or cot^-1). Take the arccot of 1.5 to find the angle.
θ = arccot(1.5)
Again, use a scientific calculator or math software to find the arccot of 1.5. The solution will be in radians. Convert it to degrees if necessary.
By using the inverse trigonometric functions mentioned above, you can find the approximate measure of all angles that satisfy the given trigonometric equations within the specified domains.