# SOLUTION: 1) Solve for m: 2 log (3m) = 2 2) Find the exact value of sin e if cot e = 5

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: 1) Solve for m: 2 log (3m) = 2 2) Find the exact value of sin e if cot e = 5      Log On

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 Question 320194: 1) Solve for m: 2 log (3m) = 2 2) Find the exact value of sin e if cot e = 5Answer 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? ======================================================================== 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? ========================================== 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?