Suppose x is a real number such that 5costx where t is a fir

Suppose x is a real number such that 5cos(t)=x where t is a first quadrant angle. Find algebraic expressions (in terms of 5 and x) for sin(t) and tan(t).

sin(t)=

tan(t)=

Solution

x is a real number such that 5cos(t)=x

5 cos(t) = x

cos(t) = x/5

then by identity sin^2(x) + cos^2(x) =1

so in \'t\' terms it is

sin^2(t) +cos^2(t) =1

sin^2(t) +(x/5)^2 =1

sin^2(t) +x^2/25 =1

sin^2(t) = 1 - x/25

sin^2(t) = (25 -x)/25

sin(t) = sqrt[ (25 -x)/25 ] [ since \'t\' is in 1st Quadrant)

sin(t) = sqrt(25-x) / 5

tan(t) = sin(t)/cos(t)

= sqrt(25-x) / 5 / x/5

= sqrt(25-x) / x