Find all the angles between 0^o and 360^o which satisfy the equation \(\sin ^{ 2 }{ \theta } =\frac { 3 }{ 4 } \)

Show that \(\sin ^{ 2 }{ \frac { \pi }{ 18 } } +\sin ^{ 2 }{ \frac { \pi }{ 9 } } +\sin ^{ 2 }{ \frac { 7\pi }{ 18 } } +\sin ^{ 2 }{ \frac { 4\pi }{ 9 } } =2\)

Show that \(\frac { sin8x\ cosx-sin6x\ cos3x }{ cos2x\ cosx-sin3x\ sin4x } =tan2x\)

Show that \(\frac { (cos\theta -cos3\theta )(sin8\theta +sin2\theta ) }{ (sin5\theta -sin\theta )(cos4\theta -cos6\theta ) } =1\)

Prove that \(\frac { sin4x+sin2x }{ cos4x+cos2x } =tan3x\)

Prove that \(\frac { cot(180^o + \theta )sin(90^o-\theta )cos(-\theta ) }{ sin(270^o+\theta )tan(-\theta )cosec(360^o+\theta ) } ={ cos }^{ 2 }\theta cot \theta \)

Find the values of other five trigonometric functions for the following
Cos \(\theta\) = -\(\frac { 1 }{ 2 },\) \(\theta\) lies in the III quadrant

In \(\triangle\)ABC, 60° prove that b + c = 2a cos \(\left( \frac { B-C }{ 2 } \right) \)

In \(\triangle\)ABC, Prove the following a sin \(\left( \frac { A }{ 2 } +B \right) \)= (b+c) sin \(\frac { A }{ 2 } \)

In \(\triangle\)ABC, Prove the following
\(\frac { asin(B-C) }{ { b }^{ 2 }-{ c }^{ 2 } } =\frac { bsin(C-A) }{ { c }^{ 2 }-{ a }^{ 2 } } =\frac { csin(A-B) }{ { a }^{ 2 }-{ b }^{ 2 } } \)

In a \(\triangle \)ABC, if \(\frac { sin \ A }{ sin \ C } =\frac { sin(A-B) }{ sin(B-c) },\), prove that a², b², c²are in arithmetic progression 

If A + B + C = \(\pi\), prove the following
i. cos A + cos B + cos C = 1 + 4 sin \(({A\over 2})\) sin \(({B\over 2})\) sin \(({C\over 2})\)
ii. sin \(({A\over 2})sin({B\over2})sin({C\over 2})\le{1\over 8}\)
iii. 1 < cos A + cos B + cos C \(\le\frac{3}{2}\)

Solve\(\sqrt{3}\)  sin \(\theta\) - cos \(\theta\) =\(\sqrt{2}\)

In a triangle  ABC, prove that \({a^2+b^2\over a^2+c^2}={1+cos(A-B)cos C\over 1+cos (A-C)cos B}\)

Suppose two radar stations located 100 km apart, each detect a fighter aircraft between them. The angle of elevation measured by the first station is 30°, whereas the angle of elevation measured by the second station is 45°. Find the altitude of the aircraft at that instant.