angle LKM is the same as angle K
angle LMK is the same as angle M
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Part (a)
sin(angle) = opposite/hypotenuse
sin(K) = ML/MK
sin(K) = 12/13
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Part (b)
cos(angle) = adjacent/hypotenuse
cos(K) = KL/MK
cos(K) = 5/13
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Part (c)
tan(angle) = opposite/adjacent
tan(K) = ML/LK
tan(K) = 12/5
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Part (d)
sin(angle) = opposite/hypotenuse
sin(M) = LK/MK
sin(M) = 5/13
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Part (e)
cos(angle) = adjacent/hypotenuse
cos(M) = ML/MK
cos(M) = 12/13
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Part (f)
tan(angle) = opposite/adjacent
tan(M) = KL/ML
tan(M) = 5/12
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Part (g)
From part (a), we found that sin(K) = 12/13, so,
sin(K) = 12/13
arcsin(sin(K)) = arcsin(12/13)
K = arcsin(12/13)
K = 67.380135
Angle LKM is approximately 67.380135 degrees
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Part (h)
Angle K and angle M are complementary. They add to 90 degrees
K+M = 90
67.380135+M = 90
67.380135+M-67.380135 = 90-67.380135
M = 22.619865
Angle LMK is roughly 22.619865 degrees