Trigonometric Ratios of Acute Angles

Rt △ ABC ∠ ACB=90° To ∠ BAC : Opposite: a=BC Hypotenuse: h=AB Adjacent: b=AC NameAbbreviationExpressionSinesina/h Cosinecosb/h Tangenttana/b Cotangentcotb/a Secantsech/b Cosecantcsch/a

sin(90°-α)=cosα cos(90°-α)=sinα tan(90°-α)=cotα cot(90°-α)=tanα sinα /cosα =tanα sin 2 α+cos 2 α=1 sinα =tanα ·cosα cosα =cotα ·sinα cotα =cosα ·cscα tanα ·cotα =1 Connections

Example 1. Rt △ ABC ∠ ACB=90° BC=6 AB=10 sin ∠ B=? cos ∠ B=? tan ∠ B=?

AC= ( )=8 AC= ( )=8 sin ∠ B= = = sin ∠ B= = = cos ∠ B= = = cos ∠ B= = = tan ∠ B= = = tan ∠ B= = = 6 10

Example 2. 0°< α <90° sin α = 0°< α <90° sin α = cos α =? cos α =?

Example 3. Rectangle ABCD, AB ＝ 10 ， BC ＝ 8 ， point E is on AD. Fold the △ CDE along CE ， point D is just on AB. Calculate the value of tan ∠ AFE.

∵ AB ＝ 10, rectangle ABCD ∴ DC=10 ∴ FC=10 ∵ FC=10,BC=8,Rt △ FCB ∴ FB=6 ∴ AF=4 If AE=x ∵ AE+ED=8, ED=EF ∴ AE+EF=8 ∴ EF=8-x ∴ x =(8-x) 2 ∴ x=3 ∴ AE=3 ∴ tan ∠ AFE=AE/AF=3/4

Square sin 2 α +cos 2 α =1 cos2 α =cos 2 α -sin 2 α =1-2sin 2 α =2cos 2 α- 1 sin2 α =2sin α cos α tan 2 α +1=1 / cos 2 α 2sin 2 α =1-cos2 α cot 2 α +1=1 / sin 2 α =1-2sin 2 α =2cos 2 α- 1 sin2 α =2sin α cos α tan 2 α +1=1 / cos 2 α 2sin 2 α =1-cos2 α cot 2 α +1=1 / sin 2 α Product sin α =tan α× cos α cos α =cot α× sin α tan α =sin α× sec α cot α =cos α× csc α sec α =tan α× csc α csc α =sec α× cot α Reciprocal tan α× cot α =1 sin α× csc α =1 cos α× sec α =1 Quotient sin α/ cos α =tan α =sec α/ csc α cos α/ sin α =cot α =csc α/ sec α

Practicing 1. △ ABC, ∠ C=90°,AB=2,AC=1,sinB=__. （ A ） 1/2 （ B ） √ 2/2 （ C ） √ 3/2 （ D ） 2 2. Rt △ ABC ，∠ C=90°, cosA=1/2, ∠ B=. 3. Rt △ ABC ，∠ C=90°, BC=a, c=___. (A)c=a  sinA (C)c=b  tanA (B)c=a/sinA (D)c=a/cosA A 30  B