Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

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

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

A. \(74^{\circ}\)

B. \(16^{\circ}\)

C. \(180^{\circ}\)

D. [tex]\(90^{\circ}\)[/tex]


Sagot :

To determine which angle correctly completes the law of cosines for the given triangle, we start with the given equation:

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

First, simplify the equation step-by-step to solve for \(\cos C\).

1. Compute the squares of each side:
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
[tex]\[ 24^2 = 576 \][/tex]

2. Substitute these values into the equation:
[tex]\[ 49 + 625 - 2(7)(25) \cos C = 576 \][/tex]

3. Simplify the left side:
[tex]\[ 674 - 2(7)(25) \cos C = 576 \][/tex]

4. Calculate the product:
[tex]\[ 2 \cdot 7 \cdot 25 = 350 \][/tex]

5. Substitute back into the equation:
[tex]\[ 674 - 350 \cos C = 576 \][/tex]

6. Isolate the term involving \(\cos C\):
[tex]\[ 674 - 576 = 350 \cos C \][/tex]
[tex]\[ 98 = 350 \cos C \][/tex]

7. Solve for \(\cos C\):
[tex]\[ \cos C = \frac{98}{350} \][/tex]
[tex]\[ \cos C = \frac{98}{350} = 0.28 \][/tex]

Next, we need to find the angle \(C\) whose cosine value is \(0.28\). This can be done by taking the arccosine (inverse cosine):

[tex]\[ C = \cos^{-1}(0.28) \][/tex]

Computing this value gives us:
[tex]\[ C \approx 73.74^\circ \][/tex]

Now, compare this value to the given options:
A. \(74^\circ\)
B. \(16^\circ\)
C. \(180^\circ\)
D. \(90^\circ\)

The closest angle to \(73.74^\circ\) is \(74^\circ\).

Therefore, the correct angle that completes the law of cosines for this triangle is:

A. [tex]\(74^\circ\)[/tex]