Trigonometry Identities

Right Triangle SOHCAHTOA Hypotenuse Side opposite to   Side adjacent to 

Examples Find exact values for all trig function in this triangle: 5 4  3

Sines, Cosines, Tangents of Special Angles

Cofunctions sin (90˚ - ) = cos  cos (90˚ - ) = sin  Ex. sin 52 ˚ = 0.7880 cos 48 ˚ = 0.7880 tan (90˚ - ) = cot  cot (90˚ - ) = tan  Ex. tan 13 ˚= 0.2309 cot 77 ˚ = 0.2309 sec (90˚ - ) = csc  csc (90˚ - ) = sec  Ex. sec 43 ˚= 1.3673 csc 47 ˚ = 1.3673

Fundamental Trig Identities Reciprocal Identities Quotient Identities

Pythagorean Identities (cos , sin ) 1 sin  cos  sin 2  + cos 2  = 1

Pythagorean Identities cos 2  + sin 2  = 1 sin 2  sin 2  sin 2  cot 2  + 1 = csc 2  cos 2  cos 2  cos 2  1 + tan 2  = sec 2 

Using Trig Identities Simplify: sin  csc  1 Simplify: tan  cos  (csc  + cot ) (csc  - cot ) csc 2  - cot 2  1

Using a Calculator Find the csc 46.89˚ The calculator does not have csc, so we must use the reciprocal identity

Applications

Applications An historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle  between the bike path and the walkway.  = 30˚

Applications Find the length of a skateboard ramp if the angle from the ground is 18.4˚ and the vertical side is 4 feet high. Ramp = 12.7 ft