Question

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

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for an angle \( I \) in triangle \( HIJ \) (where side \( i \) is opposite angle \( I \), side \( h \) is opposite angle \( H \), and side \( j \) is opposite angle \( J \)) is given by:

$$ \cos I=\frac{h^{2}+j^{2}-i^{2}}{2hj} $$

Step2: Substitute the given values

We know \( h = 46 \), \( j = 58 \), and \( i = 73 \). Substitute these into the formula:

$$ \cos I=\frac{46^{2}+58^{2}-73^{2}}{2\times46\times58} $$

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

$$ 46^{2}+58^{2}-73^{2}=2116 + 3364-5329=151 $$

Then, calculate the denominator:

$$ 2\times46\times58 = 2\times2668=5336 $$

So,

$$ \cos I=\frac{151}{5336}\approx0.0283 $$

Step3: Find the angle \( I \)

Take the arccosine of \( 0.0283 \) to find \( I \):

$$ I=\arccos(0.0283)\approx92^{\circ} $$

Answer:

b) \( 92^{\circ} \)