Math / Geometry

QuestionQuestion 13 0/1 pt 3 19
Given the triangle , find the measure of angle $A$ using the Law of Cosines. Picture is not drawn to scale $A=$ $\square$ degrees

Studdy Solution

STEP 1

What is this asking? We need to find the angle A in a triangle with sides 21, 12, and 25, where 25 is opposite to angle A. Watch out! Remember, the Law of Cosines has different forms.
Pick the one that's easiest for this problem!
Also, don't forget about inverse cosine to get the angle itself.

STEP 2

1. Set up the Law of Cosines
2. Solve for cos(A)
3. Calculate A

STEP 3

Alright, let's kick things off with the Law of Cosines!
We know that $a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(A)$ , where a, b, and c are the sides of the triangle, and A is the angle opposite side a.
In our case, $a = \textbf{25}$ , since it's opposite the angle we want to find.
Let's say $b = \textbf{21}$ and $c = \textbf{12}$ .

STEP 4

Plugging in our values, we get: $25^2 = 21^2 + 12^2 - 2 \cdot 21 \cdot 12 \cdot \cos(A)$ $625 = 441 + 144 - 504 \cdot \cos(A)$

STEP 5

Now, let's isolate $\cos(A)$ .
First, we combine the numbers on the right side: $625 = 585 - 504 \cdot \cos(A)$ Then, subtract 585 from both sides: $625 - 585 = 585 - 585 - 504 \cdot \cos(A)$ $40 = -504 \cdot \cos(A)$

STEP 6

Next, we want to get $\cos(A)$ by itself, so we divide both sides by -504: $\frac{40}{-504} = \frac{-504}{-504} \cdot \cos(A)$ $\cos(A) = -\frac{40}{504} = -\frac{\textbf{5}}{\textbf{63}}$ We've found $\cos(A)$ !

STEP 7

To find the angle A, we need to take the inverse cosine (also called arccos) of both sides: $A = \arccos\left(-\frac{5}{63}\right)$

STEP 8

Using a calculator, we find: $A \approx \textbf{94.85}^\circ$ Remember to make sure your calculator is in degree mode!

STEP 9

The measure of angle A is approximately $\textbf{94.85}$ degrees.

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QuestionQuestion 13 0/1 pt 3 19
Given the triangle , find the measure of angle $A$ using the Law of Cosines. Picture is not drawn to scale $A=$ $\square$ degrees

Studdy Solution

STEP 1

What is this asking? We need to find the angle A in a triangle with sides 21, 12, and 25, where 25 is opposite to angle A. Watch out! Remember, the Law of Cosines has different forms.
Pick the one that's easiest for this problem!
Also, don't forget about inverse cosine to get the angle itself.

STEP 2

1. Set up the Law of Cosines
2. Solve for cos(A)
3. Calculate A

STEP 3

STEP 4

Plugging in our values, we get: $25^2 = 21^2 + 12^2 - 2 \cdot 21 \cdot 12 \cdot \cos(A)$ $625 = 441 + 144 - 504 \cdot \cos(A)$

STEP 5

Now, let's isolate $\cos(A)$ .
First, we combine the numbers on the right side: $625 = 585 - 504 \cdot \cos(A)$ Then, subtract 585 from both sides: $625 - 585 = 585 - 585 - 504 \cdot \cos(A)$ $40 = -504 \cdot \cos(A)$

STEP 6

Next, we want to get $\cos(A)$ by itself, so we divide both sides by -504: $\frac{40}{-504} = \frac{-504}{-504} \cdot \cos(A)$ $\cos(A) = -\frac{40}{504} = -\frac{\textbf{5}}{\textbf{63}}$ We've found $\cos(A)$ !

STEP 7

To find the angle A, we need to take the inverse cosine (also called arccos) of both sides: $A = \arccos\left(-\frac{5}{63}\right)$

STEP 8

Using a calculator, we find: $A \approx \textbf{94.85}^\circ$ Remember to make sure your calculator is in degree mode!

STEP 9

The measure of angle A is approximately $\textbf{94.85}$ degrees.

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QuestionQuestion 13 0/1 pt 3 19Given the triangle , find the measure of angle AAA using the Law of Cosines. Picture is not drawn to scale A=A=A= □\square□ degrees

Studdy Solution

STEP 1

What is this asking? We need to find the angle *A* in a triangle with sides 21, 12, and 25, where 25 is opposite to angle *A*. Watch out! Remember, the Law of Cosines has different forms.Pick the one that's easiest for this problem!Also, don't forget about inverse cosine to get the angle itself.

STEP 2

1. Set up the Law of Cosines2. Solve for cos(*A*)3. Calculate *A*

STEP 3

STEP 4

**Plugging in** our values, we get: 252=212+122−2⋅21⋅12⋅cos⁡(A)25^2 = 21^2 + 12^2 - 2 \cdot 21 \cdot 12 \cdot \cos(A)252=212+122−2⋅21⋅12⋅cos(A) 625=441+144−504⋅cos⁡(A)625 = 441 + 144 - 504 \cdot \cos(A)625=441+144−504⋅cos(A)

STEP 5

STEP 6

STEP 7

To find the **angle** *A*, we need to take the **inverse cosine** (also called arccos) of both sides: A=arccos⁡(−563)A = \arccos\left(-\frac{5}{63}\right)A=arccos(−635​)

STEP 8

Using a calculator, we find: A≈94.85∘A \approx \textbf{94.85}^\circA≈94.85∘ Remember to make sure your calculator is in **degree mode**!

STEP 9

The measure of angle *A* is approximately 94.85\textbf{94.85}94.85 degrees.

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QuestionQuestion 13 0/1 pt 3 19
Given the triangle , find the measure of angle $A$ using the Law of Cosines. Picture is not drawn to scale $A=$ $\square$ degrees

What is this asking? We need to find the angle A in a triangle with sides 21, 12, and 25, where 25 is opposite to angle A. Watch out! Remember, the Law of Cosines has different forms.
Pick the one that's easiest for this problem!
Also, don't forget about inverse cosine to get the angle itself.

1. Set up the Law of Cosines
2. Solve for cos(A)
3. Calculate A

Plugging in our values, we get: $25^2 = 21^2 + 12^2 - 2 \cdot 21 \cdot 12 \cdot \cos(A)$ $625 = 441 + 144 - 504 \cdot \cos(A)$

To find the angle A, we need to take the inverse cosine (also called arccos) of both sides: $A = \arccos\left(-\frac{5}{63}\right)$

Using a calculator, we find: $A \approx \textbf{94.85}^\circ$ Remember to make sure your calculator is in degree mode!

The measure of angle A is approximately $\textbf{94.85}$ degrees.