To solve the equation [tex]\(5 \cos \theta + 12 \sin \theta = 13\)[/tex] and find the value of [tex]\(\tan \theta\)[/tex], let's follow these steps:
1. Rewrite and expand the equation:
We start with:
[tex]\[
5 \cos \theta + 12 \sin \theta = 13
\][/tex]
2. Square both sides of the equation:
[tex]\[
(5 \cos \theta + 12 \sin \theta)^2 = 13^2
\][/tex]
Expanding the left side, we get:
[tex]\[
(5 \cos \theta + 12 \sin \theta)^2 = 25 \cos^2 \theta + 2 \cdot 5 \cos \theta \cdot 12 \sin \theta + 144 \sin^2 \theta = 25 \cos^2 \theta + 120 \cos \theta \sin \theta + 144 \sin^2 \theta
\][/tex]
[tex]\[
25 \cos^2 \theta + 120 \cos \theta \sin \theta + 144 \sin^2 \theta = 169
\][/tex]
3. Use the Pythagorean identity [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex]:
Substitute [tex]\(\cos^2 \theta = 1 - \sin^2 \theta\)[/tex]:
[tex]\[
25 (1 - \sin^2 \theta) + 144 \sin^2 \theta + 120 \cos \theta \sin \theta = 169
\][/tex]
[tex]\[
25 - 25 \sin^2 \theta + 144 \sin^2 \theta + 120 \cos \theta \sin \theta = 169
\][/tex]
4. Simplify the equation:
Combine the [tex]\(\sin^2 \theta\)[/tex] terms:
[tex]\[
25 + 119 \sin^2 \theta + 120 \cos \theta \sin \theta = 169
\][/tex]
Subtract 25 from both sides:
[tex]\[
119 \sin^2 \theta + 120 \cos \theta \sin \theta = 144
\][/tex]
5. Express [tex]\(\cos \theta\)[/tex] in terms of [tex]\(\sin \theta\)[/tex]:
Let’s hypothesize that [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex] form a right triangle with another constant ratio, following the hint in the original expression.
6. Identify the tangent value:
Recognize that the coefficients [tex]\(5 \cos \theta\)[/tex] and [tex]\(12 \sin \theta\)[/tex] suggest a relationship consistent with a new triangle with sides representing a multiple of a known setup.
Given their relationship:
[tex]\[
5x = \cos \theta \quad \text{and} \quad 12x = \sin \theta
\][/tex]
with the hypotenuse being 13x (given by the original equation for consistency).
7. Find the tangent:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12x}{5x} = \frac{12}{5} = 2.4
\][/tex]
So, the value of [tex]\(\tan \theta\)[/tex] is:
[tex]\[
\tan \theta = 2.4
\][/tex]
