Question

If $\cos \theta =\frac{7}{25}\mathrm{and}\theta$ is in fourth quadrant, find the remaining trigonometric ratios.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{terminal}\mathrm{arm}\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{point}P(x,y). \\ \therefore r=\sqrt{x^2+y^2}...\left(i\right) \\ \mathrm{Given}:\cos \theta =\frac{7}{25}\mathrm{and}\mathrm{angle}\theta \mathrm{lies}\mathrm{in}\mathrm{quadrant}\mathrm{IV}. \\ \mathrm{We}\mathrm{know}\mathrm{that}\cos \theta =\frac{x}{r} \\ \Rightarrow \frac{x}{r}=\frac{7}{25}...\left(ii\right) \\ \mathrm{From}\left(i\right),r=\sqrt{x^2+y^2} \\ \Rightarrow r^2=x^2+y^2[\mathrm{Squaring}\mathrm{both}\mathrm{sides}] \\ \Rightarrow 1=\frac{x^2}{r^2}+\frac{y^2}{r^2}[\mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}{r}^2] \\ \Rightarrow 1={\left(\frac{7}{25}\right)}^{2^{}}+\frac{y^2}{r^2} \\ \Rightarrow \frac{y^2}{r^2}=1-{\left(\frac{7}{25}\right)}^{2^{}}=1-\frac{49}{625}=\frac{576}{625} \\ \Rightarrow \frac{y}{r}=\pm \sqrt{\frac{576}{625}}=\pm \frac{24}{25} \\ \Rightarrow \frac{y}{r}=-\frac{24}{25}.....\left(\mathrm{iii}\right)[\because \theta \mathrm{lies}\mathrm{in}\mathrm{quadrant}\mathrm{IV}] \end{gathered}$$

$$\begin{gathered} \mathrm{Since},\sin \theta =\frac{y}{r} \\ \therefore \sin \theta =-\frac{24}{25}[\theta \mathrm{lies}\mathrm{in}\mathrm{IV}\mathrm{quadrant}] \\ \mathrm{We}\mathrm{know}\mathrm{that}, \\ \tan \theta =\frac{y}{x}[\theta \mathrm{l}\mathrm{ies}\mathrm{in}\mathrm{IV}\mathrm{quadrant}] \\ \Rightarrow \tan \theta =\frac{y/r}{x/r}=\frac{-24/25}{7/25}[\mathrm{Using}(\mathrm{i})\mathrm{and}(\mathrm{ii}\left)\right] \\ \Rightarrow \tan \theta =-\frac{24}{7} \\ \mathrm{Now},\cos ec\theta =\frac{r}{y} \\ \Rightarrow \cos ec\theta =-\frac{25}{24} \\ sec\theta =\frac{r}{x} \\ \Rightarrow sec\theta =\frac{25}{7}[\mathrm{using}(ii\left)\right] \\ cot\theta =\frac{x}{y} \\ \Rightarrow cot\theta =\frac{x/r}{y/r}[\mathrm{Dividing}\mathrm{by}\mathrm{r}\mathrm{in}\mathrm{numerator}\mathrm{and}\mathrm{denominator}] \\ \Rightarrow cot\theta =\frac{7/25}{-24/25}[\mathrm{Using}(ii)and(iii\left)\right] \\ \Rightarrow cot\theta =-\frac{7}{24} \end{gathered}$$