To solve for the value of [tex]\(\tan \theta\)[/tex] given the equation [tex]\(5 \cos \theta + 12 \sin \theta = 13\)[/tex], we can perform the following steps:
1. Understand the Given Equation:
The equation provided is:
[tex]\[
5 \cos \theta + 12 \sin \theta = 13
\][/tex]
2. Square Both Sides of the Equation:
Squaring both sides will help us use the trigonometric identity that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
[tex]\[
(5 \cos \theta + 12 \sin \theta)^2 = 13^2
\][/tex]
3. Expand the Left-Hand Side using Algebra:
Expand the left-hand side of the equation:
[tex]\[
25 \cos^2 \theta + 2(5 \cos \theta)(12 \sin \theta) + 144 \sin^2 \theta = 169
\][/tex]
4. Combine Like Terms:
Notice that within the expanded form, we have terms involving [tex]\(\cos^2 \theta\)[/tex] and [tex]\(\sin^2 \theta\)[/tex]. Combine these terms:
[tex]\[
25 \cos^2 \theta + 144 \sin^2 \theta + 120 \cos \theta \sin \theta = 169
\][/tex]
5. Isolate [tex]\(\cos^2 \theta\)[/tex] and [tex]\(\sin^2 \theta\)[/tex]:
We can use the identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Let:
[tex]\[
a = 5, \quad b = 12, \quad c = 13
\][/tex]
Squaring the initial coefficients, we get:
[tex]\[
a^2 = 25, \quad b^2 = 144, \quad c^2 = 169
\][/tex]
6. Sum of Squared Terms:
From the trigonometric identity and the squared form:
[tex]\[
25 \cos^2 \theta + 144 \sin^2 \theta = 169
\][/tex]
7. Express [tex]\(\tan^2 \theta\)[/tex]:
From the equation, we obtain a ratio:
[tex]\[
\frac{144 \sin^2 \theta}{25 \cos^2 \theta} = \frac{b^2}{a^2}
\][/tex]
So:
[tex]\[
\tan^2 \theta = \frac{144}{25} = 5.76
\][/tex]
8. Find the Value of [tex]\(\tan \theta\)[/tex]:
To find [tex]\(\tan \theta\)[/tex], take the square root of both sides:
[tex]\[
\tan \theta = \sqrt{5.76} = 2.4
\][/tex]
Thus, the value of [tex]\(\tan \theta\)[/tex] is [tex]\(\boxed{2.4}\)[/tex].
