Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal plac
es where appropriate. If there is no solution, enter NO SOLUTION.) 5 sin(2θ) − 2 sin(θ) = 0
1 answer:
Answer:
x = {kπ, arccos(1/5) +2kπ, 2kπ -arccos(1/5)}
Step-by-step explanation:
The double-angle trig identity for sine is useful:
5(2sin(θ)cos(θ)) -2sin(θ) = 0
2sin(θ)(5cos(θ) -1) = 0
This has solutions that make the factors zero:
θ = arcsin(0) = kπ
and ...
cos(θ) = 1/5
θ = arccos(1/5) +2kπ . . . . or . . . . 2kπ -arccos(1/5)
_____
Some numerical values are shown on the graph attached. values for multiples of pi are ...
{..., -12.566, -9.425, -6.283, -3.142, 0, 3.142, 6.283, 9.425, 12.566, ...}
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We have the angle between the base and the hypotenuse of the triangle.
We have the height of the triangle.
We want to calculate the base of the triangle.
For this, we use the following trigonometric relationship:
Clearing x we have:
Answer:
the shortest horizontal distance that the ramp can span is:
x = 24.02 feet
We have the angle between the base and the hypotenuse of the triangle.
We have the height of the triangle.
We want to calculate the base of the triangle.
For this, we use the following trigonometric relationship:
Clearing x we have:
Answer:
the shortest horizontal distance that the ramp can span is:
x = 24.02 feet
Answer:
The series CONVERGES with the sum being 6.25.
Step-by-step explanation:
Given that the geometric series is 10 - 6 + 3.6 - 2.16 + ....
So the first term of the geometric progression is, = 10
The second term of the geometric progression is = -6 =
× r = 10 × r
where r is the constant ratio.
Therefore r = = -0.6
The third term of the geometric progression, = 3.6 =
× r = -6 × -0.6 = 3.6
The fourth term of the geometric progression, = -2.16 =
× r = 3.6 × -0.6 = -2.16
So the above confirms that the series is indeed a geometric progression with first term, = 10 and constant ratio, r = -0.6
Now to find if the series converges or diverges we can take sum of the series from first term to infinity.
We can already infer that the series is convergent since the constant ratio, r = -0.6 which is less than 1.
So from the formula for the sum to infinity of a geometric progression, S, we get
S =
So the series is convergent with the sum being 6.25