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The solutions to 1 - cos(x) = 2 - 2sin²(x) from (-π, π) are (-π/3, 0.5) and (π/3, 0.5)
<h3>How to solve the trigonometric equations?</h3><u>Equation 1: 1 - cos(x) = 2 - 2sin²(x) from (-π, π)</u>
The equation can be split as follows:
y = 1 - cos(x)
y = 2 - 2sin²(x)
Next, we plot the graph of the above equations (see graph 1)
Under the domain interval (-π, π), the curves of the equations intersect at:
(-π/3, 0.5) and (π/3, 0.5)
Hence, the solutions to 1 - cos(x) = 2 - 2sin²(x) from (-π, π) are (-π/3, 0.5) and (π/3, 0.5)
<u>Equation 2: 4cos⁴(x) - 5cos²(x) + 1 = 0 from [0, 2π)</u>
The equation can be split as follows:
y = 4cos⁴(x) - 5cos²(x) + 1
y = o
Next, we plot the graph of the above equations (see graph 2)
Under the domain interval [0, 2π), the curves of the equations intersect at:
(π/3, 0), (2π/3, 0), (π, 0), (4π/3, 0) and (5π/3, 0)
Hence, the solutions to 4cos⁴(x) - 5cos²(x) + 1 = 0 from [0, 2π) are (π/3, 0), (2π/3, 0), (π, 0), (4π/3, 0) and (5π/3, 0)
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<u>Answer:</u>
• mean = 61
• mode = 50
• median = 57.5
<u>Step-by-step explanation:</u>
• The mean is calculated by adding all the values together, and dividing the result by the number of values.
∴ mean =
⇒ mean =
⇒ mean = 61
• The mode of a set of values is the value that is the most common (has highest frequency) among them.
50 is the most common value.
∴ mode = 50
• The median is the middle-value of a set of ordered values.
∴ We have to first rearrange the set:
⇒ 40, 50, 50, 50, 55, 60, 65, 70, 80, 90
Now we need to find the middlemost value:
Since we have 10 values, which is an even number, we have to use the formula:
median =
where n is the number of values.
∴ median =
⇒ median =
The 5th and 6th terms in our ordered series are 55 and 60 respectively.
∴ median =
⇒ median = 57.5