Trigonometry

If tan⁡〖θ=8/15〗 and π≤θ≤3π/2, find the values of the other 5 trig functions.

Hello subarashiib!

\(\color{BrickRed}tan (\theta)=\frac{8}{15}\\ cot(\theta) =\color{blue}\frac{15}{8}\\ sin(\theta)=\pm \frac{tan(\theta)}{\sqrt{1+tan^2(\theta)}}=\pm \frac{\frac{8}{15}}{\sqrt{1+(\frac{8}{15})^2}}=\color{blue}\pm0.470588 \)

\(cos(\theta)=\pm\frac{1}{\sqrt{1+tan^2(\theta)}}=\pm \frac{1}{\sqrt{1+(\frac{8}{15})^2}}=\color{blue} \pm0.8823529\\ sec(\theta)=\pm\sqrt{1+tan^2(\theta)}=)=\pm\sqrt{1+(\frac{8}{15})^2}=\color{blue}\pm1.1\overline{33}\\ csc(\theta)=\pm\frac{\sqrt{1+tan^2(\theta)}}{tan(\theta)}=\pm\frac{\sqrt{1+(\frac{8}{15})^2}}{\frac{8}{15}}=\color{blue}\pm2.125\)

  !   \(\small asinus\)

Another approach:

Treat the angle as a right triangle

    Hyptotenuse = sqrt( 8^2 + 15^2 )= 17

(BUT remember this is in Quadrant III  Sin and Cos will be negative)

Tan = sin/ cos   = 8/15       sin = -8/17    cos = -15/17   cot = 1/tan = 15/8   sec = 1/cos = -17/15  csc = 1/sin = -17/8