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There are 12 pairs of integers that work.

Angle ABC is 30 degrees.

The value of m is 5.

I dont think I should do your homework for you, but I can give you a hint. 

For 1)

Rotationally use the Law of Cosines until you got one of each.

Ex: First set angle B to C, then angle A, and so on.

For 2) 

Use the law of cosines (again) to get a quadratic. The larger term is a_2, and the smaller term is a_1. (Further hint, the cosine of 60 is \(\frac{\sqrt3}{2}\))

For 3)

Use the law of cosines, where 12 is c, 10 is b, and a is a. 

side AC = 31.308

angle A = 22.619º

angle C = 45.238º

The probability is (1/4*1/3*1/2)^4*4! = 1/13824.

The graphs are tangent for k = 4.

The smallest 3 integers are: 3,  63,  123 (3 is a very valid solution). The next integer would be 183.

So, if I'm correct, (-1,0) would be the minimum point, and (0,pi/6) would be the midline point?

Thank you!~

Use the identity:  cos²(x)  =  1 - sin²(x)  so that the equation only conains the  sin  function.

cos²(x) + 3sin(x) - 3  =  0     --->     ( 1 - sin²(x) ) + 3sin(x) - 3  =  0

                                                                -sin²(x) + 3sin(x) - 2  =  0

                                                                  sin²(x) - 3sin(x) + 2  =  0

                                                            ( sin(x) - 2 )( sin(x) - 1 )  =  0

                     either     sin(x) = 2  [impossible]     or     sin(x) = 1   --->   x = pi/2

One way is to list all the possibilities for division by 4, for division by 5, and for division by 6, until you get the same number in each list.

The integers that, divided by 4, leave a remainder of 3 are:  7, 11, 15, 19, ...   (notice that these numbers increase by 4)

The integers that, divided by 5, leave a remainder of 3 are:  8, 13, 18, 23, ...   (these numbers increase by 5)

The integers that, divided by 6, leave a remainder of 3 are:  9, 15, 21, 27, ...    (these numbers increase by 6)

Extend each list until you get the same number in each list (the answer is smaller than 125).

There is a theorem in geometry that states that:  (8)(8 + 2)  =  (x)(x)

If you use the other line segment:  (4)(4 + 16)  =  (x)(x)

Either way, you end with:  80  =  x²   --->   x  =  sqrt(80)

Angle B = arctan(4 / 2) = 63.43494882 degrees

5/8 = 5+x / 10        x = 1.25

I'll let you find y    

Hint, I have an idea, what we should do is to find the length of lines by using the Pythagorean's Theory of a^2+b^2=c^2 since a lot of the incomplete triangle have 2 given side lengths.

line contains (-2,1) and (-3,3)

(y - 1)/(x + 2) = (y - 3)/(x + 3)

(y - 1)(x + 3) = (y - 3)(x + 2)

xy - x + y - 3 = xy - 3x + 2y - 6

2x - y + 3 = 0

Equation of line in standard form: 2x - y = -3

to complete the square , take 1/2 of the x coefficient and square it and add it

x^2 - 10x + 25       =    (x-5)^2

Thank you! I was able to solve it after this.

Thank you so much! That made a lot more sense with the image.

I get what ur doing but sadly the answer is incorrect. Thx for trying!

I tried those answers and they were incorrect. Do you have any idea where you went wrong?

Ah...I see, thank you. I think I prefer this way. But instead of using numbers, using variables would provide an overall proof for any angle degrees values.