Use Cosine Rule
a^(2) = b^(2) + c^(2) - 2bcCosA
Algebrically rearrange
CosA = [a^(2) - b^(2) - c^(2)] / -2bc
Substitute
CosA = [13^(2) - 12^(2) - 5^(2)# / -2(12)(5)
CosA = [ 169 - 144 - 25] / -120
Cos)A) = [0] / -120
CosA = 0
A = 90 degrees (the right angle opposite the hypotenuse)/
However,
If 'A' is the angle between '12' & '13' then 'a' is the side '5'
Hence (Notice the rearrangement of the numerical values).
CosA = [5^(2) - 12^(2) - 13^(2) ] / -2(12)(13)
CosA = [ 25 - 144 -169] / -312
CosA = [ -288[/-312
CosA = 288/312
A = Cos^(-1) [288/312]
A = 22.61986495.... degrees.
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This is a classic Pythagorean triangle. Although you have given the side lengths, you have NOT given a letter to correspond , with the given side. However, Let 12 be the adjacentr side (base) Let '5' be the opposite side ( perpendicular ) Let '13' by the hypotenuse. Sin(Angle) = opposite / hypotenuse = 5/13 Angle = Sin^(-1) 5/13 = 22.619... degrees. NB This is the angle between the hypotenuse and the base(adjacent) Now 'swopping' things around , we take the angle between the hypotenuse and the perpendicular (opposite) . This now becomes perpendicular(adjacent) and the base becomes the opposite. Hence Sin(angle) = 12/13 Angle = Sin^(-1) 12/13 = 67.380.... degrees. The angle at the 'top' of the triangle. Verification. ' 90 + 67.380... + 22.619... = 180 ( allow for calculator decimals).
Pythagoras. The right turn is assumed to be 90 degrees, and moving southwards. This forms the two shorter sides on a right angled triangle. They applying Pythagoras. h^)2) = 5^(2) + 12^(2) h^(2) = 25 + 144 h^(2) = 169 Don't forget to 'square root' both sides!!!! Hence h = sqrt(169) = 13 . This is a classic Pythagorean triangle.
13 feet
P(First card) = 13/52 P(second card = 12/51 Because you have already drawn one spade , there is one less card in the pack . P(Both spades) = 13/52 X 12/51 = 156/2652 = 0.0588
It is 3/13 - 2/13*i