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WHAT IS AN IDENTITY? A rule that can be used…. Most of the time forward as well as backwards Two functions f and g are said to be identically equal if Pre-Calculus Chapter 7 f (x) = g(x) Section 1 Basic Trigonometric Identities for all values of x for every value of x for which both functions are defined. Such an equation is referred to as an identity. An equation that is not an identity is called a conditional equation. Example ( non trig that is) x y ( x y)( x y) 2 2 They are only identities if we can prove or explain them. Prove that sin(x)tan(x) = cos(x) is not a trigonometric identity by producing a counterexample. Suppose x = /6 6 tan 6 ? ? cos 6 sin 1 2 tan 3 ? ? cos 3 2 3 not equal Go ahead try it…. Another one if needed: Prove that sin(x)cos(x) = tan(x) is not a trigonometric identity by producing a counterexample. Suppose x = /4 Notes: 1.) Producing a counterexample can show that an equation is not an identity. Proving that an equation is and identity generally takes more work. It is the showing that ALL values of the defined variable will be the challenge. 2.) I need to encourage you to start creating a careful and well organized list of all of the trigonometric identities. We will continue to accumulate this list as we continue with the trigonometry in the next trimester. At times I will allow you to use this list on quizzes and test….. Not all rules… if must be in your own handwriting and on lined paper NO example problems just the identities as given in our book or as agreed upon as a class (both classes) Suggestions: 1. Pick common angles you know from the unit circle. 2. Your basic number knowledge will help (adding, dividing, positives/ negatives, etc.) Examples 1: Prove that each equation is not a trigonometric identity by producing a counterexample A.) sec sin tan Let us try 0, B.) sec 2 1 , , , , etc. 6 4 3 2 until we find one Yes there will be other methods…. PROOF’s Sample answers a.) 4 b.) cos csc Stop and do some examples 6 1 Examples 2: use the given information to find the trigonometric value. 3 4 cos A.) if Since then find sec . Examples 2: use the given information to find the trigonometric value. B.) if 1 sec cos tan tan 2 1 sec 2 since tan then find . cos 4 cos 19 2 19 sec 4 Alternative method 3 4 cos then find 3 2 4 1 sec 19 sec 2 16 use the given information to find the trigonometric value. B.) if 3 4 Use the Pythagorean Identity 1 By substitution sec 3 4 4 So sec 3 Examples 2: tan . opp y adjacent x We can draw in the reference triangleS with those side lengths. You need to pay attention to signs and thus quadrants this reference triangle may be in. Probably 2 locations. {alright it is more of a circle than the unit circle if you really get it} Case 2 is a rotation of 180 Case 3 x axis reflection Case 4 y axis reflection Leave up on screen as you do examples on the next slide or the unit circle can address this…… Alternative method Examples 2: use the given information to find the trigonometric value. B.) if Since tan tan 3 4 , then find cos . Diagram opposite y y adjacent x x Not to scale r y -x And x2 y2 r 2 (4) 2 ( 3 ) 2 r 2 19 r 2 19 r cos Case 1 is really a rotation of 360 4 19 cos 19 4 4 19 19 19 -y x r Thus use the Pythagorean Theorem to find the missing side. Wait isn’t that what we said in the previous method??? Pythagorean Identifdy…??? some examples like page 425 example 3 Express each value as trig function on an angle in the 1 st Quadrant cos(750) 35 sin 6 cos(930) 25 tan 4 ( one more identity slide to go) 2 Explanation #1 some examples like page 425 example 3 Express each value as trig function on an angle in the 1 st Quadrant 35 sin 6 Express each value as trig function on an angle in the 1 st Quadrant cos(750) cos 30 35 sin sin 6 6 sin 30 cos(930) 25 tan tan 4 4 Some times I find it easier to draw a circle centered on the origin. Draw in the angle given. Note the quadrant you end in and the reference angle. See next slide for why/work on 35 sin 6 24 11 sin 6 6 11 sin 6 sin 6 11 6 sin 6 Since they are conterminal but you are in the wrong quadrants we need to move it into the 1st quadrant in our rewrite Flipping it over the x axis allows the sine’s to be opposites ( one more identity slide to go) Doing algebra with trig functions Express each value as trig function on an angle in the 1 st Quadrant 35 sin 6 The answer sin 6 Case 3 of the symmetric identities sin 6 6 ( one more identity slide to go) Explanation #2 using case 4 of the symmetric identities 36 sin 6 6 The answer sin(2k A) sin A Simplify each expression…. 1.) 2.) cot( ) cos( ) cos( ) sin( ) cos( ) cos( ) 1 sin( ) cos( ) csc( ) sin( ) cos( ) Quotient identities sin( ) cos( ) Symmetric identity case 2 dividing by a fraction tan( ) quotient identities Reducing your fractions ( one more identity slide to go) More Examples like those on page 426 example 4 Simplify each expression…. 3.) csc x cos x cot x 1 cos x cos x sin x sin x 1 cos 2 x sin x sin x 4.) sin x sin x cot 2 x sin x(1 cot 2 x) sin x(csc2 x) sin 2 x sin x 1 sin x 2 sin x 1 sin x sin x csc x 1 cos 2 x sin x 3