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 Trigonometry-basics/277603: Use an addition or subtraction formula to find the exact value of the expression. Tan(-5pie/12) The formula is tan(s-t)=tan s + tan t/ 1+tans tant or tan(s+t)= tan s-tan t/ 1+tans tant1 solutions Answer 202101 by nabla(475)   on 2010-03-05 19:14:40 (Show Source): You can put this solution on YOUR website!Try: To make this more natural, remember that the primitive tangent function has period of pi. So we have Tan(-5pi/12) congruent to Tan(7pi/12) mod pi. Now: What functions do we easily know the value to for trigonometrics? pi/3 is certainly one. So, pi/3=4pi/12---> We need to know 3pi/12=pi/4 ! This is also an easy find. So, we take Tan(7pi/12)=Tan(3pi/12 + 4pi/12) Now we use your formulae: Tan(3pi/12 + 4pi/12)=[Tan(pi/4)-Tan(pi/3)]/(1+Tan(pi/4)Tan(pi/3)) Now it remains to remember what Tan(pi/4) and Tan(pi/3) are. We use the definition of Tangent. Tan(x)=Sin(x)/Cos(x). So Tan(pi/4)=1. Tan(pi/3)=(sqrt(3)/2)/(1/2)=sqrt(3) Finally, substitute into our expression: (1-sqrt(3))/(1+sqrt(3)) Now, simplify: == This could be done quicker, but I hope you can see the full details of what is going on through this procedure.
 Sequences-and-series/277627: ... 23, 25, 27, 29... which expression describes the sequence? A. 3x - 3 B. 3 + 2x C. 2x - 2 D. x squared + 11 solutions Answer 202099 by nabla(475)   on 2010-03-05 18:55:04 (Show Source): You can put this solution on YOUR website!Try solving the equations for each term of the sequence. Eg. a. 3x-3=23? -> 3x=26 -> x=26/3 Hence x is not a natural number b. 3+2x=23? -> 2x=20 -> x=10 Hence this is a possible solution c. 2x-2=23? -> 2x=25 -> x=25/2 Hence x is not a natural number d. x^2+1=23 -> x^2=22 -> x is irrational and hence not a natural number. This gives (b) as the only possible solution as we define sequences such that terms are generated by natural numbers. You can easily check the remaining, given terms of the sequence.
 Geometry_Word_Problems/277639: x+7x=8x1 solutions Answer 202098 by nabla(475)   on 2010-03-05 18:40:41 (Show Source): You can put this solution on YOUR website!Perhaps it is best to think of "x" as something tangible in real life. Consider x as an orange. On the left-hand side of the equation you have 1 + 7 oranges. On the right hand side you have 8 oranges. Does 8 = 8? Yes, always. So we will say the solution is x such that x is contained in all real numbers.
 Rational-functions/197770: f(x)= 5x+4; find f(-1)1 solutions Answer 148287 by nabla(475)   on 2009-05-26 09:01:49 (Show Source): You can put this solution on YOUR website!We want to substitute -1 for x, that is x=-1... f(-1)=5(-1)+4=-5+4=-1
 Polynomials-and-rational-expressions/175746: This question is from textbook Algebra 2 How do I graph this polynomial function? f(x)=x^4 - 2x^3 - 3x^2 + 5x + 2. Thank you very much.1 solutions Answer 130891 by nabla(475)   on 2009-01-09 10:34:18 (Show Source): You can put this solution on YOUR website!1) Factor to f(x)=(-2 + x) (-1 - 3 x + x^3) 2) The first evident, real root will be x=2. 3) It should be noted that this will "eventually" resemble a parabola. Moreover, our interest will be around x=2. 4) Techniques of calculus will give us concavity, max, min, inflection. f'(x)=4x^3-6x^2-6x+5=0 gives criticals of x -> -1.05274 x -> 0.611731 x -> 1.94101 If you test outside the intervals you will note changes in direction of the slope this is indicative of extrema. Thus: x=-1.05274 is local minimum x=0.611731 is local maximum x=1.94101 is local minimum Concavity [not entirely necessary]: f''(x)=12x^2-12x-6=0 gives x -> -0.366025 x -> 1.36603 Testing intervals again you will find CU on (-inf, -0.366025), CD on (-0.366025, 1.36603), then CU on (1.36603, inf). This is certainly enough information to produce the following graph:
 expressions/175192: SIMPLIFY 30) -SR(16) = 31) SR(-16) = 32) SR(1/81) = 33) SR(25-16) = 34) SR(16-25) = 1 solutions Answer 130270 by nabla(475)   on 2009-01-06 08:03:40 (Show Source): You can put this solution on YOUR website!30. -4 31. 4i 32. 1/9 33. 3 34. 3i All of the above can be done by taking what is in the parentheses and forming a square. For instance -sqrt(16)=-sqrt(4^2)=-4.
 Polynomials-and-rational-expressions/163786: can someone please help me with this problem: Solve the equation for s. 3 - r = log (7s + 8) a. 10^3 - r - 8 / 7 b. 7 / 11 + r c. 11 + r / 7 d. 10^11 + r1 solutions Answer 120691 by nabla(475)   on 2008-10-24 11:04:47 (Show Source):
Linear-systems/163240: Solve using Cramer's Rule Solve for x and y:
x + 2y = 5
x - y = 3

1 solutions

Answer 120315 by nabla(475)   on 2008-10-21 07:37:07 (Show Source):
You can put this solution on YOUR website!
I will show you how to apply it:
for:
ax+by=e
cx+dy=f
We will have Cramer's rule giving us:

and

So, just plug in the numbers for the problem we're interested in. Namely, a=1, b=2, c=1, d=-1, e=5, f=3.
The process is done automatically by this web site:

Check:
 Solved by pluggable solver: Linear System solver (using determinant) Solve: Any system of equations: has solution or (x=3.66666666666667, y=0.666666666666667}

 Linear-equations/161716: Solve the system of equations using the substitution method. If the answer is a unique solution, present it as an ordered pair: (x,y). If not, specify whether the answer is "no solution" or "infinitely many solution." 7x+y=1 3x+2y=-9 1 solutions Answer 119182 by nabla(475)   on 2008-10-13 04:29:43 (Show Source): You can put this solution on YOUR website!from the first: y=1-7x Put that y into the second equation: 3x+2(1-7x)=-9 3x+2-14x=-9 -11x=-11 x=1 If x=1, y=1-7(1)=-6 check: 7-6=1 3-12=-9 yes.
 absolute-value/160780: How can you draw the graph of y = x* |x|? Thank you very much.1 solutions Answer 118525 by nabla(475)   on 2008-10-07 09:27:31 (Show Source): You can put this solution on YOUR website!The solution before mine is correct, but here is a more suitable graph.
 Linear-equations/160756: x+2y+z=1 2x+3y+2z=0 -x-3y+3z=11 solutions Answer 118522 by nabla(475)   on 2008-10-07 09:22:10 (Show Source): You can put this solution on YOUR website!From the first equation: z=1-x-2y Put this into the second equation: 2x+3y+2(1-x-2y)=0 2x+3y+2-2x-4y=0 -y=-2 y=2 If y=2, put that in the third equation: -x-3(2)+3(1-x-2(2))=1 -x-6+3-3x-12=1 -4x=16 x=-4 Now put y=2 and x=-4 into our z from step 1. z=1-(-4)-2(2)=1 Giving solution set {-4,2,1}. This also can be solved through techniques of linear algebra (Cramer's Rule, RREF, etc). If you need it solved in a way such as that, send me an E-mail to enabla@gmail.com .
 Vectors/160618: Find the dot product v dot w if v = i + j and w = -i + j.1 solutions Answer 118467 by nabla(475)   on 2008-10-06 16:48:23 (Show Source): You can put this solution on YOUR website!dot product means multiply each component of one vector with another vector, and add them. For instance, <1,1,1>*<0,2,3>=2+3=5 So for <1,1>*<-1,1>=0
 Polynomials-and-rational-expressions/160653: hi please help me solve 4x^2-9}}1 solutions Answer 118466 by nabla(475)   on 2008-10-06 16:44:49 (Show Source): You can put this solution on YOUR website! <- added 9 to both sides <- divided both sides by 4 x=+/- 3/2 <-took square root of both sides.
 Vectors/160619: Find b so that the vectors v = i + j and w = i + bj are orthogonal1 solutions Answer 118465 by nabla(475)   on 2008-10-06 16:41:37 (Show Source): You can put this solution on YOUR website!For v and w to be orthogonal, their dot product must be 0. (1,1)*(1,b)=1+b=0 b=-1
 Probability-and-statistics/160595: Please help! Homework question. A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-IQ society. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa?1 solutions Answer 118414 by nabla(475)   on 2008-10-06 09:50:31 (Show Source): You can put this solution on YOUR website!First we standardize the score with z=(x-u)/o where z is the distance in standard deviations from the mean, x is the score to be standardized, u is the mean of the population, and o is the standard deviation of the population [or sample]. Now, we can suspect from a percentile rank chart that the upper 2% is 2 standard deviations from the mean. In fact, however there is approximately only 100-2.27=97.73 within two standard deviations above the mean. Chances are, however, that your instructor only wants 2 standard deviations above the mean: z=(x-100)/15 15z+100=x This gives a score, x, as a function of its distance from the mean[z]. So if we want 2 standard deviations above the mean, 15(2)+100=130. You will need an IQ of 130 to get into MENSA accordingly. Strictly speaking, the score would be a little higher than 130 (in order for to be ahead of 98% of the population) due to the reasoning above.
 Percentage-and-ratio-word-problems/160137: If you earn a score of 72% on one exam and a score of 85% on another exam, what will you need to earn on the next exam so that your average for the three exams is 80%? 1 solutions Answer 118103 by nabla(475)   on 2008-10-03 09:05:09 (Show Source): You can put this solution on YOUR website!(.72+.85+x)/3=.8 1.57+x=2.4 x=.83. You need an 83%.
 Human-and-algebraic-language/160155: Write an equation for the line that passes through (-2,1) and (7,-3)1 solutions Answer 118102 by nabla(475)   on 2008-10-03 09:01:08 (Show Source): You can put this solution on YOUR website![1] Find the slope of the line connecting the two points. m=(y2-y1)/(x2-x1)=(-3-1)/(7-(-2))=-4/9. [2] Use point slope form of a line with one point: y-y1=m(x-x1). y-1=(-4/9)(x-(-2)) y=(-4/9)x+1/9
 Graphs/159919: Find the slope and the y-intercept. f(x)= -3x-10 The slope is _____ The y-intercept is (0,__)1 solutions Answer 117989 by nabla(475)   on 2008-10-02 15:37:12 (Show Source): You can put this solution on YOUR website!y=mx+b gives form with m=slope, b=y-intercept thus m=-3, b=-10
 Systems-of-equations/160013: Write an equation of the line containing the given point and perpendicular to the given line. (3,5);6x+y=2 The equation of the line is y= ____1 solutions Answer 117988 by nabla(475)   on 2008-10-02 15:31:58 (Show Source): You can put this solution on YOUR website![1] Find the slope of the given line: y=-6x+2 gives m=-6=slope [2] A perpendicular line will have slope -1/m. In this case, -1/m=1/6 [3] Apply point slope form y-y1=m(x-x1) with point (3,5)=(x1,y1). y-5=(1/6)(x-3) y=(1/6)x-1/2+10/2 y=(1/6)x+9/2. Finished
 Functions/160015: Find the slope and the y-intercept of the line. 4x=5y+151 solutions Answer 117987 by nabla(475)   on 2008-10-02 15:26:00 (Show Source): You can put this solution on YOUR website!Solve for y. (4x-15)/5=y (4/5)x-3=y The slope of a line is m in y=mx+b. b is the y-intercept [b]. Thus, we get m=4/5, y-intercept=b=-3.
 logarithm/160020: log(3x -3) - log 4 = log (x+1)1 solutions Answer 117985 by nabla(475)   on 2008-10-02 15:23:30 (Show Source): You can put this solution on YOUR website!This equation is the same as: Log((3x-3)/(4(x+1))=0 Which is moreover the same as: 4=(3x-3)/(x+1) 4x+4=3x-3 x=-7
 Inequalities/159544: b - 8 = 01 solutions Answer 117684 by nabla(475)   on 2008-09-30 12:51:18 (Show Source): You can put this solution on YOUR website!add 8 to both sides of the equation. b=8.
 Equations/159587: 3b-[2b+3a-(2a-3b)]+4a Each time I tried I got: -2b+3a Is this correct ??1 solutions Answer 117683 by nabla(475)   on 2008-09-30 12:49:21 (Show Source): You can put this solution on YOUR website!Yes, 3a-2b is the correct simplification.
 Linear-systems/159585: My problem asks me this: Solve each system of linear equations. y=x+2 y=2x Do I use the xy chart to help work this problem and where do I go from there? The problem has 2 variables and I don't know how to bring it down to one variable. Please help me!!1 solutions Answer 117682 by nabla(475)   on 2008-09-30 12:47:53 (Show Source): You can put this solution on YOUR website!You can actually just subtract the first equation from the second in this case. This gives: 0=x-2 2=x And if x=2, y=4 by back substitution. You could also solve by substitution, graphing, or other techniques from linear algebra (such as Cramer's Rule after some slight rearrangement).
 Problems-with-consecutive-odd-even-integers/159573: Find the largest of three consecutive odd integers such that the sum of the first two integers is five more than the third integer. A. 7 B. 11 C. 13 D. 91 solutions Answer 117680 by nabla(475)   on 2008-09-30 12:43:49 (Show Source): You can put this solution on YOUR website!3 consecutive odd integers: a, a+2, a+4 a+a+2=5+a+4 a=7 So our largest is 7+4=11. Check: 7+9=16 which is 5 more than 11.
 Complex_Numbers/159554: Find all roots of the equation x^3+8=0; put them in rectangular form.1 solutions Answer 117655 by nabla(475)   on 2008-09-30 09:41:47 (Show Source): You can put this solution on YOUR website!I will get you started on this problem: a^3+b^3=(a+b)(a^2-ab+b^2) and your problem here is x^3+2^3. Apply the rule. Rectangular form is simply giving the answers in (x,y).
 Numbers_Word_Problems/158838: The sum of three consecutive odd numbers is more than 375. Find the minimum values of these integers.1 solutions Answer 116994 by nabla(475)   on 2008-09-25 00:32:08 (Show Source): You can put this solution on YOUR website!3 consecutive odd numbers: a a+2 a+4 3a+6>375 3a>369 a>123 Now, if we want strictly more than 375, we will take a=125 to be our first odd number. This gives 125,127,129 as the numbers.
 Polynomials-and-rational-expressions/158851: The sum of an interger and its square is 30. What are the possible numbers?1 solutions Answer 116991 by nabla(475)   on 2008-09-25 00:28:50 (Show Source): You can put this solution on YOUR website!a+a^2=30 a^2+a-30=0 (a-5)(a+6)=0 so a=5 or a=-6 a must be an integer therefore a=-6 is excluded. a=5 is the only possible number. Check: 5+25=30. yes
 Polynomials-and-rational-expressions/158850: I am thinking of three consecutive numbers. If I multiply the first with the second and then subtract three times the third the result is 57. What are the numbers?1 solutions Answer 116989 by nabla(475)   on 2008-09-25 00:26:21 (Show Source): You can put this solution on YOUR website!3 numbers: a a+1 a+2 words to symbols: a(a+1)-3(a+2)=57 a^2+a-3a-6=57 a^2-2a-63=0 (a+7)(a-9)=0 so a=-7 or a=9. We want a to exist in the set of integers. Thus, we exclude a=-7 as a solution. The numbers, then, are 9, 10, 11. Check: 9(10)-3(11)=90-33=57 yes
 Numbers_Word_Problems/158837: Find three consecutive integers if the sum of the first two integers is 6 more than the third integer1 solutions Answer 116987 by nabla(475)   on 2008-09-25 00:23:08 (Show Source): You can put this solution on YOUR website!3 consecutive integers: a a+1 a+2 words to symbols: a+a+1=a+2+6 a=7 So the numbers are 7, 8, 9. check: first two numbers added: 15 6 more than 9: 15 yes
 Exponents-negative-and-fractional/158045: How do I solve - 1000^5/3?1 solutions Answer 116460 by nabla(475)   on 2008-09-20 15:36:17 (Show Source): You can put this solution on YOUR website!What is the third root of 1000? 10. What is 10^5? 100,000.