To find the value of [tex]\(\tan \theta\)[/tex] given the equation [tex]\(5 \cos \theta + 12 \sin \theta = 13\)[/tex], follow these steps:
1. Set up the equation and introduce new variables:
Given:
[tex]\[
5 \cos \theta + 12 \sin \theta = 13
\][/tex]
Let's denote:
[tex]\[
x = \cos \theta \quad \text{and} \quad y = \sin \theta
\][/tex]
2. Rewrite the given equation:
[tex]\[
5x + 12y = 13
\][/tex]
3. Use the Pythagorean identity:
Remember the fundamental trigonometric identity:
[tex]\[
x^2 + y^2 = 1
\][/tex]
4. Solve the system of equations:
You now have two simultaneous equations:
[tex]\[
\begin{cases}
5x + 12y = 13 \\
x^2 + y^2 = 1
\end{cases}
\][/tex]
5. Solve the first equation for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
x = \frac{13 - 12y}{5}
\][/tex]
6. Substitute this expression into the Pythagorean identity:
[tex]\[
\left( \frac{13 - 12y}{5} \right)^2 + y^2 = 1
\][/tex]
7. Solve for [tex]\(y\)[/tex]:
Simplify and solve the quadratic equation, you get:
[tex]\[
y = \frac{12}{13}
\][/tex]
8. Substitute [tex]\(y\)[/tex] back into [tex]\(x = \frac{13 - 12y}{5}\)[/tex]:
[tex]\[
x = \frac{13 - 12 \cdot \frac{12}{13}}{5} = \frac{13 - \frac{144}{13}}{5} = \frac{\frac{169 - 144}{13}}{5} = \frac{\frac{25}{13}}{5} = \frac{5}{13}
\][/tex]
So, [tex]\(\cos \theta = \frac{5}{13}\)[/tex] and [tex]\(\sin \theta = \frac{12}{13}\)[/tex].
9. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}
\][/tex]
Thus, the value of [tex]\(\tan \theta\)[/tex] is:
[tex]\[
\tan \theta = \frac{12}{5}
\][/tex]
