Question

1. in $\triangle ghi$, $g = 14$, $h = 8$, and $i = 18$. find $\cos h$ and $\sin h$. round answers to the nearest tenth.

$8^2 = 18^2 + 14^2 - 2(18)(14)\cos h$
$64 = 324 + 196 - 504\cos h$
$64 = 520 - 504\cos h$
$\cos h = \frac{456}{504}$
$h = \cos^{-1}(\frac{456}{504})$
$h = 25.2$

Explanation:

Step1: Calculate cos H using Law of Cosines

The Law of Cosines for angle \( H \) in \( \triangle GHI \) is \( h^2 = g^2 + i^2 - 2gi\cos H \). Substituting \( g = 14 \), \( h = 8 \), \( i = 18 \):
\( 8^2 = 14^2 + 18^2 - 2(14)(18)\cos H \)
\( 64 = 196 + 324 - 504\cos H \)
\( 64 = 520 - 504\cos H \)

Step2: Solve for \( \cos H \)

Subtract 520 from both sides:
\( 64 - 520 = -504\cos H \)
\( -456 = -504\cos H \)
Divide both sides by -504:
\( \cos H = \frac{456}{504} \approx 0.9 \) (rounded to nearest tenth)

Step3: Find angle \( H \) (from previous work, \( H \approx 25.2^\circ \))

Use \( \sin H = \sin(25.2^\circ) \approx 0.4 \) (rounded to nearest tenth)

Answer:

\( \cos H \approx 0.9 \), \( \sin H \approx 0.4 \)