Question

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.

Answer 1

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.

Answer 2

`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.