Answer :

GinnyAnswer
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]

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