6 tan(2x) − 6 cot(x) = 0 ; [0, 2π)
1 answer:
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Answer:
x ∈ {π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6}
Step-by-step explanation:
We can divide by 6 and use some identities to rewrite this as ...
2tan(x)/(1-tan(x)²) -1/tan(x) = 0
2tan(x)² -(1 -tan(x)²) = 0 . . . . . . . . . multiply* by tan(x)(1 -tan(x)²)
3tan(x)² = 1
tan(x) = ±1/√3 ⇒ x = {π/6, 5π/6, 7π/6, 11π/6}
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Interestingly, if we use different identities, we can rewrite it as ...
2cot(x)/(cot(x)² -1) -cot(x) = 0
cot(x)(3 -cot(x)²) = 0 . . . . . . . multiply by (cot(x)² -1)
In addition to the zeros found above, this equation also has solutions where cot(x) = 0. That is ...
x = {π/2, 3π/2}
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Usually, the first method shown would give all the solutions. Fortunately, we had a graph of the function, so we could see that the first set of solutions found was not complete. (This is one reason I like to use a graphing calculator for guidance on problems like this.)
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* In the first solution, we multiplied by the denominator tan(x)(1 -tan(x)²). It was multiplying by tan(x) that made the π/2, 3π/2 solutions disappear. Had we kept 1/tan(x) as a numerator factor of cot(x), we would have had the same result as seen in the second approach.
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Gretchen will earn $48 from mowing in 4 hours
Data
- Amount earned on each lawn = $12
- Number of hours mowing = 4 hours
This is the measure of one unit against another unit. An example of this is speed which is the rate of distance covered to time.
In this case, she mows 1 lawn in each hour and earns $12 on each.
To calculate for 4 hours
She earns $48 from mowing 4 lawns
learn more on rates here;