Join the IDNLearn.com community and start getting the answers you need today. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.
Sagot :
Let's solve for the length of the third side of a triangle given two sides and the included angle.
For a triangle with sides of lengths 2 and 3, and an included angle of [tex]\(60^\circ\)[/tex], we will use the Law of Cosines to determine the length of the third side.
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two given sides,
- [tex]\( C \)[/tex] is the included angle, and
- [tex]\( c \)[/tex] is the length of the third side.
Plugging in the given values:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( C = 60^\circ \)[/tex]
First, we convert the angle from degrees to radians because the cosine function typically requires the input angle to be in radians:
[tex]\[ 60^\circ = \frac{\pi}{3} \text{ radians} \][/tex]
[tex]\[ \cos(60^\circ) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Now, substitute these values into the Law of Cosines formula:
[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos\left(\frac{\pi}{3}\right) \][/tex]
Calculate each part step-by-step:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 2 \cdot 2 \cdot 3 \cdot \frac{1}{2} = 6 \][/tex]
Now, substitute these into the equation:
[tex]\[ c^2 = 4 + 9 - 6 \][/tex]
[tex]\[ c^2 = 13 - 6 \][/tex]
[tex]\[ c^2 = 7 \][/tex]
To find [tex]\( c \)[/tex], take the square root of both sides:
[tex]\[ c = \sqrt{7} \][/tex]
Therefore, the length of the third side is [tex]\( \sqrt{7} \)[/tex]. Thus, the correct answer is:
C. [tex]\( \sqrt{7} \)[/tex]
For a triangle with sides of lengths 2 and 3, and an included angle of [tex]\(60^\circ\)[/tex], we will use the Law of Cosines to determine the length of the third side.
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two given sides,
- [tex]\( C \)[/tex] is the included angle, and
- [tex]\( c \)[/tex] is the length of the third side.
Plugging in the given values:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( C = 60^\circ \)[/tex]
First, we convert the angle from degrees to radians because the cosine function typically requires the input angle to be in radians:
[tex]\[ 60^\circ = \frac{\pi}{3} \text{ radians} \][/tex]
[tex]\[ \cos(60^\circ) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
Now, substitute these values into the Law of Cosines formula:
[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos\left(\frac{\pi}{3}\right) \][/tex]
Calculate each part step-by-step:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 2 \cdot 2 \cdot 3 \cdot \frac{1}{2} = 6 \][/tex]
Now, substitute these into the equation:
[tex]\[ c^2 = 4 + 9 - 6 \][/tex]
[tex]\[ c^2 = 13 - 6 \][/tex]
[tex]\[ c^2 = 7 \][/tex]
To find [tex]\( c \)[/tex], take the square root of both sides:
[tex]\[ c = \sqrt{7} \][/tex]
Therefore, the length of the third side is [tex]\( \sqrt{7} \)[/tex]. Thus, the correct answer is:
C. [tex]\( \sqrt{7} \)[/tex]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.