# Chapter 11 - Section 11.3 - The Tangent Ratio and Other Ratios - Exercises - Page 519: 9

$\sin\alpha=\frac{x}{\sqrt{x^{2}+1}}$ $\cos\alpha=\frac{1}{\sqrt{x^{2}+1}}$ $\tan\alpha=x$ $\csc\alpha=\frac{\sqrt{x^{2}+1}}{x}$ $\sec\alpha=\sqrt{x^{2}+1}$ $\cot\alpha=\frac{1}{x}$

#### Work Step by Step

Let $l$ be the length of the side with an unknown length. $l=\sqrt{(\sqrt{x^{2}+1})^{2}-x^{2}}$ $l=\sqrt{x^{2}+1-x^{2}}$ $l=\sqrt{1}$ $l=1$ $\sin\alpha=\frac{opposite}{hypotenuse}=\frac{x}{\sqrt{x^{2}+1}}$ $\cos\alpha=\frac{adjacent}{hypotenuse}=\frac{1}{\sqrt{x^{2}+1}}$ $\tan\alpha=\frac{opposite}{adjacent}=\frac{x}{1}=x$ $\csc\alpha=\frac{hypotenuse}{opposite}=\frac{\sqrt{x^{2}+1}}{x}$ $\sec\alpha=\frac{hypotenuse}{adjacent}=\frac{\sqrt{x^{2}+1}}{1}=\sqrt{x^{2}+1}$ $\cot\alpha=\frac{adjacent}{opposite}=\frac{1}{x}$

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