1. find the exact value of tangent $\theta$.\nf $\frac{sqrt{5}}{2}$\ng $\frac{2}{3}$\nh $\frac{sqrt{5}}{2}$\n…

1. Find the exact value of tangent $\theta$

Explanation:

Step1: Recall tangent formula

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Given opposite side = 12 and adjacent side = 8. $\tan\theta=\frac{12}{8}=\frac{3}{2}$

Answer:

None of the given options (F, G, H, J) are correct. The value of $\tan\theta$ is $\frac{3}{2}$

2. Find the exact value of $\cos\theta$

Explanation:

Step1: Find the hypotenuse

First, find the hypotenuse $c$ using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 4$ and $b= \sqrt{8^{2}-4^{2}}=\sqrt{64 - 16}=\sqrt{48}=4\sqrt{3}$. So $c = 8$.

Step2: Recall cosine formula

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, adjacent side = $4\sqrt{3}$ and hypotenuse = 8, so $\cos\theta=\frac{4\sqrt{3}}{8}=\frac{\sqrt{3}}{2}$

Answer:

H. $\frac{\sqrt{3}}{2}$

3. Find the exact value of $\tan\theta$

Explanation:

Step1: Apply tangent formula

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Given opposite side = 30 and adjacent side = 20. $\tan\theta=\frac{30}{20}=\frac{3}{2}$

Answer:

No options provided, but the value is $\frac{3}{2}$

4. What is the augmented matrix for the given system?

Explanation:

Step1: Recall augmented matrix form

For a system of linear equations $a_1x + b_1y=c_1$ and $a_2x + b_2y=c_2$, the augmented matrix is $\left[\begin{array}{cc|c}a_1&b_1&c_1\a_2&b_2&c_2\end{array}\right]$. For the system $2x-3y=-16$ and $x + 5y=18$, the augmented matrix is $\left[\begin{array}{cc|c}2&-3&-16\1&5&18\end{array}\right]$

Answer:

C. $\left[\begin{array}{cc|c}2&-3&-16\1&5&18\end{array}\right]$