If `tantheta=1/2` , what is `sintheta` and `costheta` if `sinthetalt0` ?

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lemjay | High School Teacher | (Level 3) Senior Educator

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Take note that in order to have a positive value for tangent, the sine and cosine should have same signs.

Since the the value of sine function is less than zero, then, both sine and cosine are negative. Hence, `theta` is located at the third quadrant of polar plane.

Also, let's consider the formula of tangent which is:

`tan theta=y/x`

Since

`tan theta =1/2`

and both sine and cosine are negative, then the values of x and y of the right triangle are:

x=-2 and y =-1

Now that the legs of the right triangle are known, solve the length of the hypotenuse. To do so, apply the Pythagorean formula.

`r^2=x^2+y^2`

`r^2=(-1)^2+(-2)^2`

`r^2=1+4`

`r^2=5`

`r=+-sqrt5`

Take note that the hypotenuse of the right triangle is always positive.

Hence, the sides of the right triangle located at the third quadrant are:

`x=-2` , `y=-1` and `r=sqrt5`

Next, apply the formula of sine and cosine to get their corresponding values.

`sin theta=y/r=-1/sqrt5=-1/sqrt5*sqrt5/sqrt5=-sqrt5/5`

`cos theta=x/r=-2/sqrt5=-2/sqrt5*sqrt5/sqrt5=-(2sqrt5)/5`

Thus,   `sin theta=-sqrt5/5`    and   `cos theta=-(2sqrt5)/5` .

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