How do I solve the equation 90 cos X - 14 sin X = 72?

90, 14 and 72 are parameters as well, but those are results from another equation.

2 Answers
May 3, 2018

Write #sinx# in terms of #cosx#.

Explanation:

You can write the 14#sinx# as #14sqrt(1-cos^2 (x)# and then rearrange the terms such that the equation is:

#14sqrt(1-cos^2(x)# = 90#cosx# - 72

Upon squaring on both sides and solving the quadratic in #cosx#, you get the value of #cosx#, from which you get the value of x.

May 3, 2018

#x = 28^@97 + k360^@#
#x = - 46^@61 + k360^@#

Explanation:

90cos x - 14sin x = 72
Divide both sides by 90
#cos x - (14/90)sin x = 72/90 = 4/5 = 0.8# (1)
Call #tan t = sin t/(cos t) = 14/90 = 7/45# . Calculator gives:-->
#t = 8^@84#, and cos t = 0.988
The equation (1) becomes:
#cos x.cos t - sin t.sin x = 0.8cos t = 0.79#
Reminder: cos a.cos b - sin a.sin b = cos (a + b)
Therefor,
#cos (x + t) = 0.79 = cos 37.81#-->
#(x + t) = +- 37.81#
a. x + t = 37.81 --> #x = 37.81 - 8.84 = 28^@97#
b. x + 8.84 = - 37.81 --> #x = - 37.81 - 8.84 = - 46^@61#
For general answers, add #k360^@#
Check by calculator:
x = - 46.61 --> 90cos x = 61.83 --> 14sin x = - 10.17
90cos x - 14sin x = 61.83 + 10.17 = 72.00. Proved.