Find sin (x/2) , cos (x/2) , and tan (x/2) from the given information: cos(x) = − 7/25 , 180° < x < 270°.

Question

Find sin(x/2), cos (x/2), and tan (x/2) from the given information:

cos (x) = -7/25 180° < x < 270°

sin(x/2)=? cos (x/2)=? tan(x/2)=?

Answer 1

Hi Sofia,

For 180^o < x < 270^o we are in the quadrant III of cartesian coordinates. With cos(x) = -7/25, the hypotenuse is 25 and the x-coordinate is -7. We can find the y-coordinate by √(25² - 7²) = 24. Therefore sin(θ) = -24/25 (negative because in quad.III both sin and cos are negative).

Using the Half-Angle Formulas: (90^o < x/2 < 180^o, this puts the half angle in quad. II)

sin(x/2) = ±√[(1 - cos(x))/2] = √[(1 - (-7/25))/2] = √(16/25) = 4/5, ((+) sin, with the half angle in quad. II).

cos(x/2) = ±√[(1 - cos(x))/2] = -√[(1 + (-7/25))/2] = -√(9/25) = -3/5, (cos is negative in quad II).

tan(x/2) = [1 - cos(x)]/sin(x) = [1 - (-7/25)]/(-24/25) = -4/3, (tan is negative in quad II).

I hope this helps, Joe.