Rewriting the left hand side,
csc²t - cost sec t
= (1/sin²t)-(cost)(1/cost)
= 1/sin²t - 1
= 1/sin²t - sin²t/sin²t
= (1-sin²t)/sin²t
= cos²t/sin²t
= cot²t
It is usual to represent ratios in their simplest form so that we are not operating with large numbers. Reducing ratios to their simplest form is directly linked to equivalent fractions.
For example: On a farm there are 4 Bulls and 200 Cows. Write this as a ratio in its simplest form.
Bulls <span>: </span>Cows
4 <span>: </span>200
If we halve the number of bulls then we must halve the number of cows so that the relationship between the bulls and cows stays constant. This gives us:
Bulls <span>: </span>Cows
2 <span>: </span>100
Halving again gives us
1 <span>: </span>50
So the ratio of Bulls to Cows equals 1 : 50. The ratio is now represented in its simplest form.
An example where we have 3 quantities.
On the farm there are 24 ducks, 36 geese and 48 hens.
Ratio of ducks <span>: </span>geese <span>: </span>hens
24 <span>: </span>36 <span>: </span>48
Dividing each quantity by 12 gives us
2 <span>: </span>3 : 4
So the ratio of ducks to geese to hens equals 2 : 3 : 4 which is the simplest form since we can find no further common factor.
C because the x-axis and the y-axis are switched, basically.
Parallelogram as well as a quadrilateral with pairs of equal and parallel opposite sides
Given:
Two similar triangles AEB and ACD.
To solve for x, we can use ratio and proportion following the rule for similar triangles.
So,
AE / AD = AB / AC
Substitute values,
AE = AD - DE
AE = 6 - 5
AE = 1
1 / 6 = 2 / (2 + x + 4)
Solve for x
x = 6
then,
BC = x +4 = 6 + 4 = 10
AC = 2 + x + 4 = 12
Therefore, the measurements of the triangles are:
AE = 1
BC = 10
AC = 12
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