Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine which of the given equations correctly uses the law of cosines to solve for [tex]\( y \)[/tex], we need to carefully examine each equation and ensure it aligns with the standard form of the law of cosines.
The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
In this context, we can match:
- [tex]\( a \)[/tex] with [tex]\( y \)[/tex], which is the side we need to solve for.
- [tex]\( b \)[/tex] with [tex]\( 9 \)[/tex]
- [tex]\( c \)[/tex] with [tex]\( 19 \)[/tex]
- [tex]\( A \)[/tex] with [tex]\( 41^\circ \)[/tex], which is the angle opposite side [tex]\( y \)[/tex].
Let's rewrite each option to see if it matches the form [tex]\( a^2 = b^2 + c^2 - 2bc \cos(A) \)[/tex].
1. [tex]\( 9^2 = y^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because in the law of cosines, [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] should all be on one side of the equation. Here, the equation has the variables mixed up across both sides.
2. [tex]\( y^2 = 9^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect. The term [tex]\( 2(y)(19) \cos(41^\circ) \)[/tex] should involve the sides we know the lengths of, not the side we are solving for (i.e., it should involve 9 and 19, not [tex]\(y\)[/tex]).
3. [tex]\( 9^2 = y^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because it incorrectly places [tex]\( y^2 \)[/tex] with the known side lengths [tex]\( 9 \)[/tex] and [tex]\( 19 \)[/tex] on the right-hand side of the equation.
4. [tex]\( y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation correctly follows the structure of the law of cosines:
[tex]\[ y^2 = 9^2 + 19^2 - 2 \cdot 9 \cdot 19 \cdot \cos(41^\circ) \][/tex]
This is in the form:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
where [tex]\( a = y \)[/tex], [tex]\( b = 9 \)[/tex], [tex]\( c = 19 \)[/tex], and [tex]\( A = 41^\circ \)[/tex].
Thus, the correct equation that uses the law of cosines to solve for [tex]\( y \)[/tex] is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
So, the correct answer is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
In this context, we can match:
- [tex]\( a \)[/tex] with [tex]\( y \)[/tex], which is the side we need to solve for.
- [tex]\( b \)[/tex] with [tex]\( 9 \)[/tex]
- [tex]\( c \)[/tex] with [tex]\( 19 \)[/tex]
- [tex]\( A \)[/tex] with [tex]\( 41^\circ \)[/tex], which is the angle opposite side [tex]\( y \)[/tex].
Let's rewrite each option to see if it matches the form [tex]\( a^2 = b^2 + c^2 - 2bc \cos(A) \)[/tex].
1. [tex]\( 9^2 = y^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because in the law of cosines, [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] should all be on one side of the equation. Here, the equation has the variables mixed up across both sides.
2. [tex]\( y^2 = 9^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect. The term [tex]\( 2(y)(19) \cos(41^\circ) \)[/tex] should involve the sides we know the lengths of, not the side we are solving for (i.e., it should involve 9 and 19, not [tex]\(y\)[/tex]).
3. [tex]\( 9^2 = y^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because it incorrectly places [tex]\( y^2 \)[/tex] with the known side lengths [tex]\( 9 \)[/tex] and [tex]\( 19 \)[/tex] on the right-hand side of the equation.
4. [tex]\( y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation correctly follows the structure of the law of cosines:
[tex]\[ y^2 = 9^2 + 19^2 - 2 \cdot 9 \cdot 19 \cdot \cos(41^\circ) \][/tex]
This is in the form:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
where [tex]\( a = y \)[/tex], [tex]\( b = 9 \)[/tex], [tex]\( c = 19 \)[/tex], and [tex]\( A = 41^\circ \)[/tex].
Thus, the correct equation that uses the law of cosines to solve for [tex]\( y \)[/tex] is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
So, the correct answer is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.