set equal to zero

Question

6 (sec(x))(tan(x)) - 9 (cot(x))(csc(x)) = 0

I know the answer is .85 from a calculator, but how would I go about solving this?

Answer 1

6secxtanx -9cotxcscx= 0

if not sure what to do, try converting all trig functions to sines and cosines

although there may be easier ways to do this problem

6(1/cosx)(sinx/cosx) = 9(cosx/sinx)(1/sinx)

6/9= (cosx/sinx)

2/3= (cosx/sin^2(x))/(sinx/cos^2(x))

2/3 = (cosx/sin^2(x))(cos^2(x)/sinx)

2/3 = cos^3(x)/sin^3(x) = cot^3(x)

cotx = cube root of 2/3 = about .87

x = cot^-1(.87) = about 49 degrees = about .85 radians

check the answer:

6sec49tan49 - 9cot49csc49

= 6(1.52)(1.15)- 9(.87)(1.33)

= about 10.5 - 10.4 = about 0

close enough, given the rounding approximations

or if you have a graphing calculator

graph it and see where the curve intesects the x axis

you'll get an infinite number of interesections,

but you apparently want just the quadrant I solution