Analytic Trigonometry 5.1 FUNdamental Identities Haha, FUNdamental. Get it? Fundamental…FUN…nm.

Our primary applications of trigonometry so far have been computational. We have not made full use of the properties of the functions to study the connections among the trigonometric functions themselves. We will now shift our emphasis more toward theory and proof, exploring where the properties of these special functions lead us, often with no immediate concern for real-world relevance at all. Hopefully in the process you will gain an appreciation for the rich and intricate tapestry of interlocking patterns that can be woven from the six basic trig functions – patterns that will take on even greater beauty later on when you can view them through the lens of calculus.

5.1 Fundamental Identities During today’s lesson you will learn: Definition of an identity Basic Trigonometric Identities, Pythagorean Identities, Cofunction and Even-Odd Identities Simplify Trigonometric Expressions and Solve Trigonometric Equations

Which represents an identity? 1 + 1 = 2 2(x – 3) = 2x – 6 x2 + 3 = 7 (x2 – 1)/(x+1) = x – 1 What are the similarities and differences in each of the equations above?

What is an identity? Identities - statements which are true for all values of the variable for which both sides of the equation are defined In other words, an identity is an equation that is ALWAYS equal for values which are appropriate for its domain. Domain of validity – set of values for which an equation is defined.

Basic Trigonometric Identities Reciprocal Identities Quotient Identities

Exploration 1 Domain of Validity Complete the exploration on p. 445 6 min. 1st 4 min. – NO TALKING!! Last 2 min. – You can discuss with your neighbor Write answers on your own paper!!

Exploration 2 Pythagorean Theorem Use your calculator to evaluate each expression Write Value What conclusions can you draw about sin2q + cos2q ? (sin(25))2 + (cos(25))2 (sin(72))2 + (cos(72))2 (sin(90))2 + (cos(90))2 (sin(30))2 + (cos(30))2

Pythagorean Identities sin2 q + cos 2 q = 1 1 + tan2 q = sec 2 q cot 2 q + 1 = csc 2 q How can you change the first identity into the second? The third?

Ex 1 Without a calculator Find sin q & cos q if tan q = 8 & sin q > 0.

Exploration 3 Cofunction Identities Type in as shown on your calculator Compare Evaluate What conclusions can you draw about sin (90 - q) & cos q? cos (90 - q) & sin q? tan (90 - q) & cot q? cot (90 - q) & tan q? sin (90 - 65) cos (65) cos (90 - 71) sin (71) tan (90 - 68) cot (68) cot (90 - 47) tan (47)

Cofunction Identities

Exploration 4 Odd-Even Identities Evaluate Compare S or O Which functions are even? Which functions are odd? sin (-25) sin (25) cos (-25) cos (25) tan (-25) tan (25) cot (-25) cot (25) csc (-25) csc (25) sec (-25) sec (25)

Odd-Even Identities

Ex 2 If sin ( - p/2) = 0.73, find cos (-). Ex 3 If cot(-) = 7.89 , find tan ( - p/2).

Simplifying Trigonometric Expressions Ex 4 Use basic identities to simplify the expression. a) 1 + tan2x b) sec2 (-x) – tan2x csc2 x

Ex 5 Use basic identities to simplify the expression. (sin x) (tan x + cot x) tan x + tan x csc2 x sec2 x

Solving Trigonometric Equations Ex 6 Find all values of x in the interval [0,2p) that solve 2 cos x sin x – cos x = 0. Algebraically Verify by graphing

Ex 6 Find all solutions to the equation in the interval [0,2p) Ex 6 Find all solutions to the equation in the interval [0,2p). You do NOT need a calculator! 4 cos2 x – 4 cos x + 1 = 0 2 sin2 x + 3 sin x = 2

Ex 7 Use your calculator to solve cos x = 0.75 sin2x = 0.4

Exiting Thoughts… Mrs. Mullen asked the class to factor 1 – sin2x. Fundamental Identities Exiting Thoughts… Mrs. Mullen asked the class to factor 1 – sin2x. Ana wrote (1 – sin x)(1 + sin x). Cara wrote (cos x)(cos x). Who is correct? Explain how you made your choice.