MHB Solving for $\theta$ and $H$ in Trig Equation

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To solve for the unknowns \(H\) and \(\theta\) in the equations \(H\cos(\theta)+559.68=750\) and \(H\sin(\theta)-124.26=0\), the system can be rewritten as \(H\cos(\theta)=190.32\) and \(H\sin(\theta)=124.26\). By squaring and adding these equations, \(H\) can be determined as \(H=\sqrt{(190.32)^2+(124.26)^2}\), resulting in \(H=227.29\). The angle \(\theta\) is then calculated using \(H\cos(\theta)=190.32\), yielding \(\theta=\cos^{-1}(\frac{190.32}{227.29})\), which is approximately \(33.14^{\circ}\). The discussion concludes with a question about whether \(H\) must be positive.
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I need help finding the unknowns

$H\cos(\theta)+559.68=750$
$H\sin(\theta)-124.26=0$
 
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I need help finding the unknowns

I know I have two unknowns and two equations but sin cos made me confuse when I try to solve one of the variable and plug it in one of the equations please help. Thanks!

$H\cos(\theta)+559.68=750$
$H\sin(\theta)-124.26=0$
 
Hello. :) What have you tried?
 
I would first write the system in the form:

$$H\cos(\theta)=a$$

$$H\sin(\theta)=b$$

Square and add both equations, and you can eliminate $\theta$. What do you find?

edit: I have merged the two duplicate threads.
 
MarkFL said:
I would first write the system in the form:

$$H\cos(\theta)=a$$

$$H\sin(\theta)=b$$

Square and add both equations, and you can eliminate $\theta$. What do you find?

edit: I have merged the two duplicate threads.

Or to avoid extraneous solutions, you can divide the second equation by the first, thereby making an equation just in terms of $\displaystyle \begin{align*} \tan{(\theta )} \end{align*}$...
 
Hi MarkFL!

$\displaystyle H\cos(\theta)=a$(1)
$\displaystyle H\sin(\theta)=b$(2)

solvind for a and b

$a=190.32$
$b=124.26$

squaring and adding both equations I have

$H^2(\sin^2(\theta)+\cos^2(\theta))=(190.32)^2+(124.26)^2$
$H^2(1)= (190.32)^2+(124.26)^2$
$H=\sqrt((190.32)^2+(124.26)^2$
$H=227.29$

Solving for theta using equation (1) or (2)

Using (1)

$H\cos(\theta)=a$
$227.29\cos(\theta)=190.32$

$\theta=\cos^{-1}(\frac{190.32}{227.29})$
$\theta=33.14^{\circ}$

:))) hooray!
 
Does $H$ have to be positive?
 

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