# Six Trigonometric Ratios (sin, cos, tan, cot, sec and cosec)

Trigonometric Ratios (sin, cos, tan, cot, sec and cosec)

These six trigonometric ratios form the base of trigonometry. So, learn them carefully. Lets suppose we have triangle ABC right angled at B. We have angle $\theta$ and $\alpha$ in $\triangle ABC$.

Note that opposite side of angle $\theta$ is AB and opposite side of angle $\alpha$ is BC.

Adjacent side to angle $\theta$ is BC and adjacent side to angle $\alpha$ is AB.

AC is the hypotenuse of triangle ABC.

Now you have to memorize this sentence "Old Harry And His Old Aunt". Don't worry it is not an English class. It will help you to memorize formulas of six trigonometric ratios which are sin, cos, tan, sec, cosec and cot. Now look at all the capital letters of the sentence which are O, H, A, H, O and A. Now suppose that O stands for opposite side, H for hypotenuse and A for adjacent side. First two capital letters form sin, next two form cos and last two form tan. Confused? 🙁 Don't worry it will be clear with the following.

$sin \theta=\frac{O}{H}=\frac{opposide side}{hypotenuse}=\frac{AB}{AC}$

$cos \theta=\frac{A}{H}=\frac{adjacent side}{hypotenuse}=\frac{BC}{AC}$

$tan \theta=\frac{O}{A}=\frac{opposite side}{adjacent side}=\frac{AB}{BC}$

Similarly,

$sin \alpha=\frac{O}{H}=\frac{opposide side}{hypotenuse}=\frac{BC}{AC}$

$cos \alpha=\frac{A}{H}=\frac{adjacent side}{hypotenuse}=\frac{AB}{AC}$

$tan \alpha=\frac{O}{A}=\frac{opposite side}{adjacent side}=\frac{BC}{AB}$

Note that cosec is inverse of sin, sec is inverse of cos and cot is inverse of tan.

Therefore,

$cosec \alpha=\frac{1}{sin\alpha}=\frac{AC}{BC}$

$sec \alpha=\frac{1}{cos\alpha}=\frac{AC}{AB}$

$cot\alpha=\frac{1}{tan\alpha}=\frac{AB}{BC}$

Similarly,

$cosec \theta=\frac{1}{sin\theta}=\frac{AC}{AB}$

$sec \theta=\frac{1}{cos\theta}=\frac{AC}{BC}$

$cot\theta=\frac{1}{tan\theta}=\frac{BC}{AB}$

Trigonometric Ratios 1

Trigonometric Ratios 2

Posted in : Trigonometry

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