Question

match the following trigonometric ratios with the correct equation for the triangle shown below.
\\( cot a= )
\\( csc a= )
\\( sec a= )
\\( \tan a= )
\\( cos a= )
\\( sin a= )
\\( cot c= )
\\( csc c= )
\\( sec c= )
\\( \tan c= )
\\( cos c= )
\\( sin c= )

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, for an acute angle $\theta$: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$, $\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}$, $\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}$. For angle $A$ in right - triangle $ABC$ with right - angle at $B$, the opposite side to angle $A$ is $BC = 24$, the adjacent side is $AB = 7$, and the hypotenuse is $AC = 25$.

Step2: Calculate trigonometric ratios for angle $A$

$\sin A=\frac{BC}{AC}=\frac{24}{25}$, $\cos A=\frac{AB}{AC}=\frac{7}{25}$, $\tan A=\frac{BC}{AB}=\frac{24}{7}$, $\cot A=\frac{AB}{BC}=\frac{7}{24}$, $\sec A=\frac{AC}{AB}=\frac{25}{7}$, $\csc A=\frac{AC}{BC}=\frac{25}{24}$.

Step3: Calculate trigonometric ratios for angle $C$

For angle $C$, the opposite side is $AB = 7$, the adjacent side is $BC = 24$, and the hypotenuse is $AC = 25$. So, $\sin C=\frac{AB}{AC}=\frac{7}{25}$, $\cos C=\frac{BC}{AC}=\frac{24}{25}$, $\tan C=\frac{AB}{BC}=\frac{7}{24}$, $\cot C=\frac{BC}{AB}=\frac{24}{7}$, $\sec C=\frac{AC}{BC}=\frac{25}{24}$, $\csc C=\frac{AC}{AB}=\frac{25}{7}$.

Answer:

$\sin A=\frac{24}{25}$
$\cos A=\frac{7}{25}$
$\tan A=\frac{24}{7}$
$\cot A=\frac{7}{24}$
$\sec A=\frac{25}{7}$
$\csc A=\frac{25}{24}$
$\sin C=\frac{7}{25}$
$\cos C=\frac{24}{25}$
$\tan C=\frac{7}{24}$
$\cot C=\frac{24}{7}$
$\sec C=\frac{25}{24}$
$\csc C=\frac{25}{7}$