#### Question

Suppose there exists a point P(x,3) \, (x \ne 0) on the terminal side of an angle \theta with its vertex at the origin, and \cos \theta = \dfrac{\sqrt{10} }{10}x . Show the value of \sin \theta and \tan \theta

Question

Suppose there exists a point P(x,3) \, (x \ne 0) on the terminal side of an angle \theta with its vertex at the origin, and \cos \theta = \dfrac{\sqrt{10} }{10}x . Show the value of \sin \theta and \tan \theta

According to the definition of conse

\cos \theta = \dfrac{x}{\sqrt{x^2+9} } = \dfrac{\sqrt{10} }{10}x

\because x \ne 0

\dfrac{1}{\sqrt{x^2+9} } = \dfrac{\sqrt{10} }{10}

Square both sides and simplify,

\dfrac{1}{x^2+9}=\dfrac{1}{10}

Solve the equation

x=1 or x= -1

If x = 1, the coordinate of P is (1,3)

\sin \theta = \dfrac{3}{\sqrt{10} }=\dfrac{3\sqrt{10} }{10} , \tan \theta = 3

If x = -1, the coordinate of P is (-1,3),

\sin \theta = \dfrac{3}{\sqrt{10} }=\dfrac{3\sqrt{10} }{10} , \tan \theta = -3