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Select the angle that correctly completes the law of cosines for this triangle.

[tex]\[ 24^2 + 25^2 - 2(24)(25) \cos \theta = 7^2 \][/tex]

A. [tex]\( \theta = 45^\circ \)[/tex]
B. [tex]\( \theta = 60^\circ \)[/tex]
C. [tex]\( \theta = 90^\circ \)[/tex]
D. [tex]\( \theta = 120^\circ \)[/tex]



Answer :

GinnyAnswer
To find the angle that correctly completes the law of cosines equation for the given triangle, we start with the given side lengths:

[tex]\( a = 24 \)[/tex]
[tex]\( b = 25 \)[/tex]
[tex]\( c = 7 \)[/tex]

The law of cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]

Substituting the given values:
[tex]\[ 7^2 = 24^2 + 25^2 - 2(24)(25) \cos(C) \][/tex]

Simplifying:
[tex]\[ 49 = 576 + 625 - 2(24)(25) \cos(C) \][/tex]

Combining the constants:
[tex]\[ 49 = 1201 - 1200 \cos(C) \][/tex]

Rearranging to solve for [tex]\( \cos(C) \)[/tex]:
[tex]\[ 1200 \cos(C) = 1201 - 49 \][/tex]

[tex]\[ 1200 \cos(C) = 1152 \][/tex]

[tex]\[ \cos(C) = \frac{1152}{1200} \][/tex]

[tex]\[ \cos(C) = 0.96 \][/tex]

Now, we need to find the angle [tex]\(C\)[/tex] whose cosine is 0.96. We use the inverse cosine function:
[tex]\[ C = \arccos(0.96) \][/tex]

Calculating the angle in degrees:
[tex]\[ C \approx 16.26^\circ \][/tex]

So, the angle that completes the law of cosines for this triangle is approximately [tex]\(16.26^\circ\)[/tex].

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