Precalculus (6th Edition) Blitzer

The length of the side is $b=7$ and the values of the six trigonometric functions are $\sin \theta =\frac{24}{25}$ , $\cos \theta =\frac{7}{25}$ , $\tan \theta =\frac{24}{7}$ , $\csc \theta =\frac{25}{24}$ , $\sec \theta =\frac{25}{7}$ and $\cot \theta =\frac{7}{24}$.
In the right angle triangle, $a=24$ , $b$ is the side adjacent with angle $\theta$ and $c=25$. According to the Pythagoras theorem, ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$ Rearrange for $b$. \begin{align} & {{b}^{2}}={{c}^{2}}-{{a}^{2}} \\ & b=\sqrt{{{c}^{2}}-{{a}^{2}}} \\ \end{align} Substitute $25$ for $c$ and $24$ for $a$. \begin{align} & b=\sqrt{{{\left( 25 \right)}^{2}}-{{\left( 24 \right)}^{2}}} \\ & =\sqrt{625-576} \\ & =\sqrt{49} \\ & =7 \end{align} The ratio of $\sin \theta$ is $\sin \theta =\frac{a}{c}$ Substitute $25$ for $c$ and $24$ for $a$. $\sin \theta =\frac{24}{25}$ The ratio of $\cos \theta$ is $\cos \theta =\frac{b}{c}$ Substitute $25$ for $c$ and $7$ for $b$. $\cos \theta =\frac{7}{25}$ The ratio of $\tan \theta$ is $\tan \theta =\frac{a}{b}$ Substitute $24$ for $a$ and $7$ for $b$. $\tan \theta =\frac{24}{7}$ The ratio of $\csc \theta$ is $\csc \theta =\frac{c}{a}$ Substitute $25$ for $c$ and $24$ for $a$. $\csc \theta =\frac{25}{24}$ The ratio of $\sec \theta$ is $\sec \theta =\frac{c}{b}$ Substitute $25$ for $c$ and $7$ for $b$. $\sec \theta =\frac{25}{7}$ The ratio of $\cot \theta$ is $\cot \theta =\frac{b}{a}$ Substitute $24$ for $a$ and $7$ for $b$. $\cot \theta =\frac{7}{24}$ Therefore, the length of the side is $b=7$ and the values of the six trigonometric functions are, $\sin \theta =\frac{24}{25}$ , $\cos \theta =\frac{7}{25}$ , $\tan \theta =\frac{24}{7}$ , $\csc \theta =\frac{25}{24}$ , $\sec \theta =\frac{25}{7}$ and $\cot \theta =\frac{7}{24}$.