Fundamental Trigonometric Identities MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011
Objectives In this lesson we will learn to: recognize and write the fundamental trigonometric identities, use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and to rewrite trigonometric expressions.
Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u
Quotient and Reciprocal Identities tan u = sin u cos u sin u = 1 csc u cos u = 1 sec u tan u = 1 cot u cot u = cos u sin u csc u = 1 sin u sec u = 1 cos u cot u = 1 tan u
Cofunction Identities ( π ) sin 2 u ( π ) cos 2 u ( π ) tan 2 u = cos u = sin u = cot u ( π ) csc 2 u ( π ) sec 2 u ( π ) cot 2 u = sec u = csc u = tan u
Even/Odd Identities sin ( u) = sin u cos( u) = cos u tan( u) = tan u csc ( u) = csc u sec( u) = sec u cot( u) = cot u
Example (1 of 9) If csc θ = 5 and cos θ < 0 find the values of all six trigonometric functions.
Example (1 of 9) If csc θ = 5 and cos θ < 0 find the values of all six trigonometric functions. sin θ = cosθ = tanθ = cotθ = secθ = csc θ = 5
Example (1 of 9) If csc θ = 5 and cos θ < 0 find the values of all six trigonometric functions. sin θ = 1 5 cosθ = 2 6 5 6 tanθ = 12 cotθ = 2 6 secθ = 5 6 12 csc θ = 5
Example (2 of 9) Factor and simplify the following expression. sin 2 x csc 2 x sin 2 x =
Example (2 of 9) Factor and simplify the following expression. sin 2 x csc 2 x sin 2 x = sin 2 x(csc 2 x 1) = sin 2 x(cot 2 x) ( cos = sin 2 2 ) x x sin 2 x = cos 2 x
Example (3 of 9) Factor and simplify the following expression. sec 4 x tan 4 x =
Example (3 of 9) Factor and simplify the following expression. sec 4 x tan 4 x = (sec 2 x + tan 2 x)(sec 2 x tan 2 x) = (sec 2 x + tan 2 x)(1) = sec 2 x + tan 2 x
Example (4 of 9) Factor and simplify the following expression. cos 4 x 2 cos 2 x + 1 =
Example (4 of 9) Factor and simplify the following expression. cos 4 x 2 cos 2 x + 1 = (cos 2 x 1) 2 = ( sin 2 x) = sin 4 x
Example (5 of 9) Carry out the multiplication and simplify the following expression. (cot x + csc x)(cot x csc x) =
Example (5 of 9) Carry out the multiplication and simplify the following expression. (cot x + csc x)(cot x csc x) = cot 2 x csc 2 x = (csc 2 x cot 2 x) = 1
Example (6 of 9) Perform the subtraction and simplify the following expression. 1 sec x + 1 1 sec x 1 =
Example (6 of 9) Perform the subtraction and simplify the following expression. 1 sec x + 1 1 sec x 1 sec x 1 = (sec x + 1)(sec x 1) sec x + 1 (sec x 1)(sec x + 1) sec x 1 (sec x + 1) = sec 2 x 1 2 = tan 2 x = 2 cot 2 x
Example (7 of 9) Rewrite the following expression so that it is not in fractional form. 5 sec x + tan x =
Example (7 of 9) Rewrite the following expression so that it is not in fractional form. 5 sec x + tan x 5 (sec x tan x) = (sec x + tan x) (sec x tan x) 5(sec x tan x) = (sec x + tan x)(sec x tan x) 5(sec x tan x) = sec 2 x tan 2 x 5(sec x tan x) = 1 = 5 sec x 5 tan x
Example (8 of 9) Substitute x = 2 cosθ with 0 < θ < π/2 in the expression 64 16x 2 and simplify.
Example (8 of 9) Substitute x = 2 cosθ with 0 < θ < π/2 in the expression 64 16x 2 and simplify. 64 16x 2 = = = = 64 16(2 cosθ) 2 64 64 cos 2 θ 64(1 cos 2 θ) 64 sin 2 θ = 8 sin θ since sin θ > 0 when 0 < θ < π/2.
Example (9 of 9) Rewrite the following expression as a single logarithm and simplify the result. ln tan x + ln csc x =
Example (9 of 9) Rewrite the following expression as a single logarithm and simplify the result. ln tan x + ln csc x = ln tan x csc x = ln sin x 1 cos x sin x = ln 1 cos x = ln sec x
Homework Read Section 5.1. Exercises: 1, 5, 9, 13,..., 113, 117