Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Sure! Let's go through the solution step-by-step using the Law of Cosines.
Given:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( C = 45^\circ \)[/tex] (the angle in degrees)
We need to find the length of side [tex]\( c \)[/tex], which is opposite the given angle [tex]\( C \)[/tex].
### Step 1: Convert angle [tex]\( C \)[/tex] to radians
The Law of Cosines requires the angle in radians. To convert degrees to radians, we use the formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
So,
[tex]\[ C_{\text{radians}} = 45^\circ \times \frac{\pi}{180} \approx 0.7854 \text{ radians} \][/tex]
### Step 2: Apply the Law of Cosines
The Law of Cosines formula is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Substituting the given values:
[tex]\[ a = 5 \][/tex]
[tex]\[ b = 7 \][/tex]
[tex]\[ \cos(C_{\text{radians}}) = \cos(0.7854) \][/tex]
We need the cosine of [tex]\( 0.7854 \)[/tex]. Using a calculator, we find:
[tex]\[ \cos(0.7854) \approx 0.7071 \][/tex]
Thus, plugging in the values:
[tex]\[ c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot 0.7071 \][/tex]
Calculating each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 2 \cdot 5 \cdot 7 \cdot 0.7071 \approx 49.4975 \][/tex]
Putting it all together:
[tex]\[ c^2 = 25 + 49 - 49.4975 \][/tex]
[tex]\[ c^2 \approx 24.5025 \][/tex]
### Step 3: Solve for [tex]\( c \)[/tex]
Now, solve for [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{24.5025} \][/tex]
[tex]\[ c \approx 4.95 \][/tex]
### Step 4: Round to the nearest tenth
Finally, rounding [tex]\( 4.95 \)[/tex] to the nearest tenth:
[tex]\[ c \approx 5.0 \][/tex]
Therefore, the length of side [tex]\( c \)[/tex] is approximately [tex]\( 5.0 \)[/tex] units.
Given:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( C = 45^\circ \)[/tex] (the angle in degrees)
We need to find the length of side [tex]\( c \)[/tex], which is opposite the given angle [tex]\( C \)[/tex].
### Step 1: Convert angle [tex]\( C \)[/tex] to radians
The Law of Cosines requires the angle in radians. To convert degrees to radians, we use the formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
So,
[tex]\[ C_{\text{radians}} = 45^\circ \times \frac{\pi}{180} \approx 0.7854 \text{ radians} \][/tex]
### Step 2: Apply the Law of Cosines
The Law of Cosines formula is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Substituting the given values:
[tex]\[ a = 5 \][/tex]
[tex]\[ b = 7 \][/tex]
[tex]\[ \cos(C_{\text{radians}}) = \cos(0.7854) \][/tex]
We need the cosine of [tex]\( 0.7854 \)[/tex]. Using a calculator, we find:
[tex]\[ \cos(0.7854) \approx 0.7071 \][/tex]
Thus, plugging in the values:
[tex]\[ c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot 0.7071 \][/tex]
Calculating each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 2 \cdot 5 \cdot 7 \cdot 0.7071 \approx 49.4975 \][/tex]
Putting it all together:
[tex]\[ c^2 = 25 + 49 - 49.4975 \][/tex]
[tex]\[ c^2 \approx 24.5025 \][/tex]
### Step 3: Solve for [tex]\( c \)[/tex]
Now, solve for [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{24.5025} \][/tex]
[tex]\[ c \approx 4.95 \][/tex]
### Step 4: Round to the nearest tenth
Finally, rounding [tex]\( 4.95 \)[/tex] to the nearest tenth:
[tex]\[ c \approx 5.0 \][/tex]
Therefore, the length of side [tex]\( c \)[/tex] is approximately [tex]\( 5.0 \)[/tex] units.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.