Question 320194: 1) Solve for m: 2 log (3m) = 2
2) Find the exact value of sin e if cot e = 5
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! For question 1, we use the fact that y = log_b (x) if and only if b^(y) = x.
Let b = 10^2
Let x = (3m)^2
We now have the following equation:
10^2 = (3m)^2
100 = 9m^2
100/9 = m^2
11.11111111 = m^2
Take square root of both sides.
sqrt{11.11111111} = sqrt{m^2}
3.3333333333 = m
The decimal number 3.333333333 can be written as 10/3 as the value of m.
Did you follow?
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For the trig question, the first thing to notice is that sine and cotagent are both positive, which means we are in quadrant 1.
We are in quadrant 1 using a right triangle to find the hypotenuse via the famous Pythagorean Theorem.
cot(e) = 5 = 5/1
sin(e) = 1/(hyp)
Also keep in mind:
cotangent = adj/opp
sine = opp/hyp
We need to find the hypotenuse of the right triangle formed in quadrant 1.
1^2 + 5^2 = (hyp)^2
26 = (hyp)^2
We now take the square root of both sides.
sqrt{26} = sqrt{(hyp)^2}
sqrt{26} = hypotenuse
Are you with me so far?
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Since we now know the value of the hypotenuse, we can plug that into
sin(e) = 1/(hyp), which becomes sin(e) = 1/sqrt{26}.
It is never a good idea to leave a radical in the denominator of a fraction.
We now need to rationalize the denominator. This is done to remove the radical from the denominator.
To do so, multiply the TOP and BOTTOM by the sqrt{26}.
sin(e) = 1/sqrt{26} times [sqrt{26}/sqrt{26}]
sin(e) = sqrt{26}/26
Did you follow?
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