Sure, let's solve these problems step by step.
### Problem 7
Given the equation [tex]\( 4 \tan \theta = 3 \)[/tex], we need to show that:
[tex]\[
\frac{4 \sin \theta + 3 \cos \theta}{8 \sin \theta + 5 \cos \theta} = \frac{6}{11}
\][/tex]
First, solve for [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{3}{4}
\][/tex]
Now, recall that:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
From [tex]\(\tan \theta = \frac{3}{4}\)[/tex], we can assume:
[tex]\[
\sin \theta = 3k \quad \text{and} \quad \cos \theta = 4k \quad \text{for some constant } k
\][/tex]
Now, substitute [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] into the expression:
[tex]\[
\frac{4 \sin \theta + 3 \cos \theta}{8 \sin \theta + 5 \cos \theta} = \frac{4(3k) + 3(4k)}{8(3k) + 5(4k)}
\][/tex]
Simplify the numerator and denominator:
[tex]\[
= \frac{12k + 12k}{24k + 20k} = \frac{24k}{44k}
\][/tex]
The [tex]\(k\)[/tex] terms cancel out:
[tex]\[
= \frac{24}{44} = \frac{6}{11}
\][/tex]
Thus, we have shown that:
[tex]\[
\frac{4 \sin \theta + 3 \cos \theta}{8 \sin \theta + 5 \cos \theta} = \frac{6}{11}
\][/tex]
### Problem 8
Given [tex]\( \cos \theta = \frac{\sqrt{3}}{2} \)[/tex], we need to show that:
[tex]\[
4 \cos^3 \theta - 3 \cos \theta = 0
\][/tex]
First, calculate [tex]\( \cos^3 \theta \)[/tex]:
[tex]\[
\cos^3 \theta = \left( \frac{\sqrt{3}}{2} \right)^3 = \frac{(\sqrt{3})^3}{2^3} = \frac{3\sqrt{3}}{8}
\][/tex]
Now, substitute [tex]\(\cos \theta\)[/tex] and [tex]\(\cos^3 \theta\)[/tex] into the expression:
[tex]\[
4 \cos^3 \theta - 3 \cos \theta = 4 \left( \frac{3\sqrt{3}}{8} \right) - 3 \left( \frac{\sqrt{3}}{2} \right)
\][/tex]
Simplify the expression:
[tex]\[
= \frac{12\sqrt{3}}{8} - \frac{3\sqrt{3}}{2}
\][/tex]
Convert the terms to have a common denominator:
[tex]\[
= \frac{12\sqrt{3}}{8} - \frac{12\sqrt{3}}{8}
\][/tex]
Subtract the fractions:
[tex]\[
= \frac{12\sqrt{3} - 12\sqrt{3}}{8} = \frac{0}{8} = 0
\][/tex]
Thus, we have shown that:
[tex]\[
4 \cos^3 \theta - 3 \cos \theta = 0
\][/tex]
These solutions verify the given mathematical statements.
