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### Part b: Find the smallest angle in the triangle with sides ratio [tex]\(2:3:4\)[/tex]
To find the angles of a triangle with side lengths in the ratio [tex]\(2:3:4\)[/tex], we use the Law of Cosines, which is stated as:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Considering the sides [tex]\(a = 2\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 4\)[/tex]:
1. Calculate angle [tex]\(A\)[/tex] opposite to side [tex]\(a = 2\)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{3^2 + 4^2 - 2^2}{2 \cdot 3 \cdot 4} = \frac{9 + 16 - 4}{24} = \frac{21}{24} = \frac{7}{8} \][/tex]
[tex]\[ A = \cos^{-1}\left(\frac{7}{8}\right) \approx 28.96^\circ \][/tex]
2. Calculate angle [tex]\(B\)[/tex] opposite to side [tex]\(b = 3\)[/tex]:
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{2^2 + 4^2 - 3^2}{2 \cdot 2 \cdot 4} = \frac{4 + 16 - 9}{16} = \frac{11}{16} \][/tex]
[tex]\[ B = \cos^{-1}\left(\frac{11}{16}\right) \approx 46.57^\circ \][/tex]
3. Calculate angle [tex]\(C\)[/tex] opposite to side [tex]\(c = 4\)[/tex]:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{2^2 + 3^2 - 4^2}{2 \cdot 2 \cdot 3} = \frac{4 + 9 - 16}{12} = \frac{-3}{12} = -\frac{1}{4} \][/tex]
[tex]\[ C = \cos^{-1}\left(-\frac{1}{4}\right) \approx 104.48^\circ \][/tex]
Among the calculated angles:
- [tex]\(A \approx 28.96^\circ\)[/tex]
- [tex]\(B \approx 46.57^\circ\)[/tex]
- [tex]\(C \approx 104.48^\circ\)[/tex]
The smallest angle is [tex]\(A \approx 28.96^\circ\)[/tex].
### Part c: Algebraic expressions
i. Factorize completely [tex]\(2xy - 6y + 7x - 21\)[/tex]
Group the terms for easier factoring:
[tex]\[ 2xy - 6y + 7x - 21 \][/tex]
Group the terms:
[tex]\[ 2y(x - 3) + 7(x - 3) \][/tex]
Factor out the common factor [tex]\((x - 3)\)[/tex]:
[tex]\[ = (x - 3)(2y + 7) \][/tex]
Thus, the completely factored form is:
[tex]\[ 2xy - 6y + 7x - 21 = (x - 3)(2y + 7) \][/tex]
ii. Evaluate the expression in (i) if [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex]
Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] into [tex]\((x - 3)(2y + 7)\)[/tex]:
[tex]\[ = (2 - 3)(2(-1) + 7) \][/tex]
[tex]\[ = (-1)(-2 + 7) \][/tex]
[tex]\[ = (-1)(5) \][/tex]
[tex]\[ = -5 \][/tex]
So, the value of the expression when [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] is [tex]\(-5\)[/tex].
In summary:
1. The smallest angle in the triangle with sides ratio 2:3:4 is approximately [tex]\(28.96^\circ\)[/tex].
2. The completely factored form of [tex]\(2xy - 6y + 7x - 21\)[/tex] is [tex]\((x - 3)(2y + 7)\)[/tex].
3. The value of the expression when [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] is [tex]\(-5\)[/tex].
### Part b: Find the smallest angle in the triangle with sides ratio [tex]\(2:3:4\)[/tex]
To find the angles of a triangle with side lengths in the ratio [tex]\(2:3:4\)[/tex], we use the Law of Cosines, which is stated as:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]
Considering the sides [tex]\(a = 2\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 4\)[/tex]:
1. Calculate angle [tex]\(A\)[/tex] opposite to side [tex]\(a = 2\)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{3^2 + 4^2 - 2^2}{2 \cdot 3 \cdot 4} = \frac{9 + 16 - 4}{24} = \frac{21}{24} = \frac{7}{8} \][/tex]
[tex]\[ A = \cos^{-1}\left(\frac{7}{8}\right) \approx 28.96^\circ \][/tex]
2. Calculate angle [tex]\(B\)[/tex] opposite to side [tex]\(b = 3\)[/tex]:
[tex]\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{2^2 + 4^2 - 3^2}{2 \cdot 2 \cdot 4} = \frac{4 + 16 - 9}{16} = \frac{11}{16} \][/tex]
[tex]\[ B = \cos^{-1}\left(\frac{11}{16}\right) \approx 46.57^\circ \][/tex]
3. Calculate angle [tex]\(C\)[/tex] opposite to side [tex]\(c = 4\)[/tex]:
[tex]\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{2^2 + 3^2 - 4^2}{2 \cdot 2 \cdot 3} = \frac{4 + 9 - 16}{12} = \frac{-3}{12} = -\frac{1}{4} \][/tex]
[tex]\[ C = \cos^{-1}\left(-\frac{1}{4}\right) \approx 104.48^\circ \][/tex]
Among the calculated angles:
- [tex]\(A \approx 28.96^\circ\)[/tex]
- [tex]\(B \approx 46.57^\circ\)[/tex]
- [tex]\(C \approx 104.48^\circ\)[/tex]
The smallest angle is [tex]\(A \approx 28.96^\circ\)[/tex].
### Part c: Algebraic expressions
i. Factorize completely [tex]\(2xy - 6y + 7x - 21\)[/tex]
Group the terms for easier factoring:
[tex]\[ 2xy - 6y + 7x - 21 \][/tex]
Group the terms:
[tex]\[ 2y(x - 3) + 7(x - 3) \][/tex]
Factor out the common factor [tex]\((x - 3)\)[/tex]:
[tex]\[ = (x - 3)(2y + 7) \][/tex]
Thus, the completely factored form is:
[tex]\[ 2xy - 6y + 7x - 21 = (x - 3)(2y + 7) \][/tex]
ii. Evaluate the expression in (i) if [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex]
Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] into [tex]\((x - 3)(2y + 7)\)[/tex]:
[tex]\[ = (2 - 3)(2(-1) + 7) \][/tex]
[tex]\[ = (-1)(-2 + 7) \][/tex]
[tex]\[ = (-1)(5) \][/tex]
[tex]\[ = -5 \][/tex]
So, the value of the expression when [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] is [tex]\(-5\)[/tex].
In summary:
1. The smallest angle in the triangle with sides ratio 2:3:4 is approximately [tex]\(28.96^\circ\)[/tex].
2. The completely factored form of [tex]\(2xy - 6y + 7x - 21\)[/tex] is [tex]\((x - 3)(2y + 7)\)[/tex].
3. The value of the expression when [tex]\(x = 2\)[/tex] and [tex]\(y = -1\)[/tex] is [tex]\(-5\)[/tex].
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