91Ó°ÊÓ

If $\mathrm{θ}$is any angle in a given right angle triangle, then the three trignometric ratios are,

$\begin{array}{c}sin\mathrm{θ}=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰o±è±è´Ç²õ¾±³Ù±ð â¶Ä‰t´Ç â¶Ä‰}∠\mathrm{θ}\text{ â¶Ä‰â¶Ä‰}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ \\ cos\mathrm{θ}=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰a»åÂá²¹³¦±ð²Ô³Ù â¶Ä‰t´Ç â¶Ä‰}∠\mathrm{θ}\text{ â¶Ä‰â¶Ä‰}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ \\ tan\mathrm{θ}=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰o±è±è´Ç²õ¾±³Ù±ð â¶Ä‰t´Ç â¶Ä‰}∠\mathrm{θ}}{\text{±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰a»åÂá²¹³¦±ð²Ô³Ù â¶Ä‰t´Ç â¶Ä‰}∠\mathrm{θ}\text{ â¶Ä‰}}\end{array}$

Observe the figure.

From the figure, $AB$is hypotenuse (side opposite to 90 degree) and length of $AB$is 5 units.

$CB$is the side opposite to angle $A$and length of $CB$is 4 units.

$AC$is the side adjacent to angle $A$and length of $AC$is 3 units.

$\begin{array}{c}sinA=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰o±è±è´Ç²õ¾±³Ù±ð â¶Ä‰t´Ç â¶Ä‰}∠A\text{ â¶Ä‰â¶Ä‰}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ =\frac{\text{length â¶Ä‰of â¶Ä‰CB}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ =\frac{4}{5}\\ \\ cosA=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰a»åÂá²¹³¦±ð²Ô³Ù â¶Ä‰t´Ç â¶Ä‰}∠A\text{ â¶Ä‰â¶Ä‰}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ =\frac{\text{length â¶Ä‰of â¶Ä‰AC}}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰h²â±è´Ç³Ù±ð²Ô³Ü²õ±ð}}\\ =\frac{3}{5}\\ \\ tanA=\frac{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰o±è±è´Ç²õ¾±³Ù±ð â¶Ä‰t´Ç â¶Ä‰}∠A}{\text{±ô±ð²Ô²µ³Ù³ó â¶Ä‰o´Ú â¶Ä‰s¾±»å±ð â¶Ä‰a»åÂá²¹³¦±ð²Ô³Ù â¶Ä‰t´Ç â¶Ä‰}∠A\text{ â¶Ä‰}}\\ =\frac{\text{length â¶Ä‰of â¶Ä‰CB}}{\text{length â¶Ä‰of â¶Ä‰AC}}\\ =\frac{4}{3}\end{array}$

The three trigonometric ratios are $\sin A$,$\cos A$, $\tan A$and their values are as follows.

$\begin{array}{c}\sin A=\frac{4}{5}\\ \cos A=\frac{3}{5}\\ \tan A=\frac{4}{3}\end{array}$