Question

In a \[\Delta ABC\], \[a = 5,b = 4\] and \[cos\left( {A - B} \right) = 3132\] , then side c is equal to
A. \[6\]
B. \[7\]
C. \[9\]
D. None of these

Answer 1

Hint
The hint states that side c is equal to 6. Side c can be found by taking the inverse cosine of \[AB - BC\]. This will give you \[AC\], which when multiplied with \[3132\] gives us side c as \[6\]. Arccosine is another name for inverse cosine. It is the cos function's opposite. Also known as "arccos," sometimes. When two sides of a right triangle's length are known, it is used to measure the unknown angle.
Cosine cannot be a one-to-one function; hence its domain must be constrained to the range 0 to pi. This function is known as the restricted cosine function. \[Cos - 1\left( x \right)\] or arccos is the notation for the inverse cosine function \[\left( x \right)\].
The sine function's inverse is arcsine. It is used to calculate the sine value of an angle, which is the ratio of the opposite side to the hypotenuse. Therefore, we may determine the angle's measurement if we are aware of the lengths of the opposing side and hypotenuse.
Complete step-by-step solution
The equation is,
\[tan(\frac{{A - B}}{2}) = \sqrt {\frac{{1 - \cos (A - B)}}{{1 + \cos (A - B)}}} \]
\[\sqrt {\frac{{1 - (31/32)}}{{1 + (31/32)}}} \]
Then, the equation becomes,
 => \[\frac{1}{{\sqrt {63} }}\]
\[\frac{{a - b}}{{a + b}}\cot \frac{C}{2} = \frac{1}{{\sqrt {63} }}\]
So, after solving the equation it becomes
\[\frac{1}{9}\cot \frac{C}{2} = \frac{1}{{\sqrt {63} }}\]
\[ = > \tan \frac{C}{2} = \frac{{\sqrt 7 }}{3}\]
\[\cos C = \frac{{1 - {{\tan }^2}(C/2)}}{{1 + {{\tan }^2}(C/2)}}\]
The values are being substituted in the equation,
\[\cos C = \frac{{1 - (7/9)}}{{1 + (7/9)}} = \frac{1}{8}\]
\[{c^2} = {a^2} + {b^2} - 2ab\cos C\]
\[{c^2} = 25 + 16 - 40 \times \frac{1}{8} = 36\]
\[ = > c = 6\]
_{So, option A is correct.}
note
The side c of \[\Delta ABC\] is equal to \[6\]. This can be found by solving for \[sin\left( x \right)\] in the equation \[cos\left( {A - B} \right) = 3132\], where A and B are the coordinates of side c.
The reverse to determine a point's coordinates in a coordinate system. Start at the point, then move up or down a vertical line until you reach the x-axis. Your x-coordinate is shown there. To get the y-coordinate, repeat the previous step while adhering to a horizontal line.
An ordered pair of numbers, known as an x-coordinate and a y-coordinate respectively, can be used to represent each point in a graph. Parentheses are used to indicate ordered pairings (x-coordinate, y-coordinate). The starting point is at \[\left( {0,0} \right)\].