Question

What is the Trigonometric value for sin θ = -1/3, 180º < θ < 270º ; tan θ?

Answer 1

`tan(theta)=1/(2sqrt(2)`

Explanation:

The pythagorean trigonmetric identity states

`cos^2(x)+sin^2(x)=1=>cos(x)=sqrt(1-sin^2(x))`

For `sin(theta)=-1/3` cosine must be

`cos(theta)=+-sqrt(1-(-1/3)^3)=+-sqrt(1-1/9)=+-sqrt(8)/3`

When `180^@< theta <270^@` the angle must be in the 3. quadrant, therefore cosine must be negative

`cos(theta)=-sqrt(8)/3`

By the definition of tangent

`tan(theta)=sin(theta)/cos(theta)=(-1/3)/(-sqrt(8)/3)=1/sqrt(8)=1/(2sqrt(2))`

Answer 2

`tantheta=sqrt2/4`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)tantheta=sintheta/costheta`

`•color(white)(x)sin^2theta+cos^2theta=1`

`rArrcostheta=+-sqrt(1-sin^2theta)`

`"since "180^@< theta< 270^@`

`"then "costheta<0" and "tantheta>0`

`rArrcostheta=-sqrt(1-(-1/3)^2)`

`color(white)(rArrcostheta)=-sqrt(1-1/9)=-sqrt(8/9)=-(2sqrt2)/3`

`rArrtantheta=(-1/3)/(-(2sqrt2)/3)=1/(2sqrt2)xxsqrt2/sqrt2=sqrt2/4`