Fundamental Trigonometric Identities

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1 Fundamental Trigonometric Identities MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 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.

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

4 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

5 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

6 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

7 Example (1 of 9) If csc θ = 5 and cos θ < 0 find the values of all six trigonometric functions.

8 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

9 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 = tanθ = 12 cotθ = 2 6 secθ = csc θ = 5

10 Example (2 of 9) Factor and simplify the following expression. sin 2 x csc 2 x sin 2 x =

11 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

12 Example (3 of 9) Factor and simplify the following expression. sec 4 x tan 4 x =

13 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

14 Example (4 of 9) Factor and simplify the following expression. cos 4 x 2 cos 2 x + 1 =

15 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

16 Example (5 of 9) Carry out the multiplication and simplify the following expression. (cot x + csc x)(cot x csc x) =

17 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

18 Example (6 of 9) Perform the subtraction and simplify the following expression. 1 sec x sec x 1 =

19 Example (6 of 9) Perform the subtraction and simplify the following expression. 1 sec x 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

20 Example (7 of 9) Rewrite the following expression so that it is not in fractional form. 5 sec x + tan x =

21 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

22 Example (8 of 9) Substitute x = 2 cosθ with 0 < θ < π/2 in the expression 64 16x 2 and simplify.

23 Example (8 of 9) Substitute x = 2 cosθ with 0 < θ < π/2 in the expression 64 16x 2 and simplify x 2 = = = = 64 16(2 cosθ) cos 2 θ 64(1 cos 2 θ) 64 sin 2 θ = 8 sin θ since sin θ > 0 when 0 < θ < π/2.

24 Example (9 of 9) Rewrite the following expression as a single logarithm and simplify the result. ln tan x + ln csc x =

25 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

26 Homework Read Section 5.1. Exercises: 1, 5, 9, 13,..., 113, 117

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas