Question

for triangle hij, if h = 46, i = 73, and j = 58, use the cosine law to solve for \\( \angle i \\). round your answer to the nearest degree.
a) \\( 88^\circ \\)
b) \\( 92^\circ \\)
c) \\( 53^\circ \\)
d) \\( 39^\circ \\)

Explanation:

Step1: Recall Cosine Law Formula

The cosine law for finding an angle \( \theta \) (let's say angle \( I \)) in a triangle with sides \( h \), \( j \), and \( i \) opposite to angles \( H \), \( J \), and \( I \) respectively is:
\( \cos I=\frac{h^{2}+j^{2}-i^{2}}{2hj} \)

Step2: Substitute the Given Values

Given \( h = 46 \), \( j = 58 \), \( i = 73 \). Substitute into the formula:
\( \cos I=\frac{46^{2}+58^{2}-73^{2}}{2\times46\times58} \)

First, calculate the numerator:
\( 46^{2}=2116 \), \( 58^{2}=3364 \), \( 73^{2}=5329 \)
\( 2116 + 3364-5329=2116 + 3364=5480; 5480 - 5329 = 151 \)

Then, calculate the denominator:
\( 2\times46\times58 = 2\times2668 = 5336 \)

So, \( \cos I=\frac{151}{5336}\approx0.0283 \)

Step3: Find the Angle

Take the arccosine of \( 0.0283 \):
\( I=\arccos(0.0283)\approx88^{\circ} \) (rounded to the nearest degree)

Answer:

a) \( 88^\circ \)