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Wade Arellano
Mathematics
29 December, 13:24
Solve tan x=sin x giving all possible solutions
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Lorelai Colon
29 December, 13:53
0
x = 2 π n_1 for n_1 element Z
or x = π n_2 for n_2 element Z
Step-by-step explanation:
Solve for x:
tan (x) = sin (x)
Subtract sin (x) from both sides:
tan (x) - sin (x) = 0
Factor sin (x) from the left hand side:
sin (x) (sec (x) - 1) = 0
Split sin (x) (sec (x) - 1) into separate parts with additional assumptions.
Assume cos (x) !=0 from sec (x):
sec (x) - 1 = 0 or sin (x) = 0 for cos (x) !=0
Add 1 to both sides:
sec (x) = 1 or sin (x) = 0 for cos (x) !=0
Take the reciprocal of both sides:
cos (x) = 1 or sin (x) = 0 for cos (x) !=0
Take the inverse cosine of both sides:
x = 2 π n_1 for n_1 element Z
or sin (x) = 0 for cos (x) !=0
Take the inverse sine of both sides:
x = 2 π n_1 for n_1 element Z
or x = π n_2 for cos (x) !=0 and n_2 element Z
The roots x = π n_2 never violate cos (x) !=0, which means this assumption can be omitted:
Answer: x = 2 π n_1 for n_1 element Z
or x = π n_2 for n_2 element Z
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