Given that sintheatre=5/13,find tan(90-theatre)in it's simplest form
First, we need to find the value of cos(theatre) using the identity: sin^2(theatre) + cos^2(theatre) = 1.
Since sin(theatre) = 5/13, we can square it to get sin^2(theatre):
sin^2(theatre) = (5/13)^2 = 25/169.
Now, we can subtract sin^2(theatre) from 1 to find cos^2(theatre):
cos^2(theatre) = 1 - 25/169 = 144/169.
Taking the square root of both sides, we find:
cos(theatre) = √(144/169) = 12/13.
Now, we can use the tangent identity: tan(90 - theatre) = sin(theatre) / cos(theatre).
tan(90 - theatre) = sin(theatre) / cos(theatre) = (5/13) / (12/13) = 5/12.
Therefore, tan(90 - theatre) simplifies to 5/12.
To find tan(90-theatre), we can use the following trigonometric identity:
tan(90 - theatre) = cot(theatre)
Since sin(theatre) = 5/13, we can find cos(theatre) using the Pythagorean identity:
sin^2(theatre) + cos^2(theatre) = 1
(5/13)^2 + cos^2(theatre) = 1
25/169 + cos^2(theatre) = 1
cos^2(theatre) = 1 - 25/169
cos^2(theatre) = 169/169 - 25/169
cos^2(theatre) = 144/169
cos(theatre) = ± √(144/169)
Since cos(theatre) is positive and theatre lies in the first quadrant, cos(theatre) = √(144/169) = 12/13.
Now let's find cot(theatre) using the definition:
cot(theatre) = cos(theatre) / sin(theatre)
cot(theatre) = (12/13) / (5/13)
cot(theatre) = (12/13) * (13/5)
cot(theatre) = 12/5
Therefore, tan(90 - theatre) = cot(theatre) = 12/5.