Question

If in a triangle the angles are in A. P. and \[b:c = \sqrt 3 :\sqrt 2 \], then \[\angle A\] is equal to
A. \[30^\circ \]
B. \[60^\circ \]
C. \[15^\circ \]
D. \[75^\circ \]

Answer 1

Hint: It is possible to compute the length of a side in the right triangle using slope-intercept form given that it knows the intercepts and slopes.
Using this information, the process to solve for Side Using algebraic manipulation. To solve this problem, it is helpful to use the Law of Sines. The Law of Sines states that in a right triangle, the sine of one angle is equal to the sum of the sine of other two angles. In this case, \[\angle A = sin\left( {75^\circ } \right)\]and \[sin\left( {50^\circ } \right) = cos\left( {75^\circ } \right)\]. Therefore, \[\angle A\]is \[75\] degree.

Complete step by step solution: The given equation is \[b:c = \sqrt 3 :\sqrt 2 \]
The objective is to find \[\angle A\]
Let the equation be \[\angle A = a - d\], \[\angle B = a\]
And \[\angle C = a + d\]
The sum of angles of triangle can be written as,
\[a - d + a + a + d = 180^\circ \]
This equation can also be written as
\[3a = 180^\circ \]
\[a = 60^\circ \]
Hence, \[\angle B = 60^\circ \]
Use the sine law in the equation,
\[\dfrac{{\sin B}}{b} = \dfrac{{\sin C}}{c}\]
This equation is rewritten as,
\[\dfrac{{\sin 60^\circ }}{{\sin c}} = \dfrac{b}{c}\]
The value of \[\sin 60^\circ \]is \[\sqrt 3 /2\]
So. With the value, the equation is written as,
\[\dfrac{{\sqrt 3 /2}}{{\sin c}} = \dfrac{{\sqrt 3 }}{{\sqrt 2 }}\]
\[\sin c = \dfrac{1}{{\sqrt 2 }}\]
The value of \[\dfrac{1}{{\sqrt 2 }}\]is \[\sin 45^\circ \]
\[\sin c = \sin 45^\circ \]
As, \[\angle C\]is equal to \[45^\circ \]
As we already know \[\angle C = a + d\]
\[a + d = 45^\circ \]
\[d = 45^\circ - 60^\circ = - 15^\circ \]
 \[\angle A = a - d = 60^\circ - ( - 15^\circ )\]
\[\angle A = 75^\circ \]

So, the correct option is D that is \[75^\circ \].

Note: The computational geometry tool known as Graphing Calculator. a correlation between a triangle's opposite side and the angle's size. When we know about one angle and the side opposite it, as well as another angle and the side opposite it, we may apply the sine rule to determine the missing angle or side in a triangle.
After entering the information into our calculator, we can see that \[\angle A\]is equal to\[75^\circ \].