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TrigonometryApplying ASTC, Reference, Coterminal Angles In the old days, when people didnt have a calculator and only had a table of reference for the trigonometric function values of angles from 0° to 90°, its crucial to know how to express the function in terms of a positive acute angle (i.e., its reference angle). Express as a function of a positive acute angle. 1.sin 123° = 2.cos 234° = 3.tan –345° = SATCSATC 4.sin 456° = 5.cos 12,345° = 6.tan –6,789° = How to find the sine, cosine, tangent of special angles without a calculator: A special angle is an angle that are multiples of 30, 45 and 60. Examples: 1. sin 120 =2. cos 135 = 3. tan 225 = 4. sin 210 = 5. cos 300 = 6. tan 330 = 45° 60° 30° Page 15

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TrigonometrySecant, Cosecant and Cotangent The Other Three Functions We have learned the 3 basic definitionssine, cosine and tangent. There are 3 more definitionssecant (sec), cosecant (csc) and cotangent (cot). Adjacent leg Opposite leg Hypotenuse The basic three:The other three: It turns out most people dont remember the definitions of secant, cosecant and cotangent, instead, they remember them as the ____________ of the basic three. Q:How can we memorize the other three as the reciprocals the basic three without mixing them up? A:Just remember they pair each other with a different initial letter: tan sec csc Page 16

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SATCSATC If is in Q I, we know: sin = (+), cos = (+), tan = (+). What about... csc = ( ), sec = ( ), cot = ( ) If is in Q II, we know: sin = (+), cos = (–), tan = (–). What about... csc = ( ), sec = ( ), cot = ( ) If is in Q III, we know: sin = (–), cos = (–), tan = (+). What about... csc = ( ), sec = ( ), cot = ( ) If is in Q IV, we know: sin = (–), cos = (+), tan = (–). What about... csc = ( ), sec = ( ), cot = ( ) How to use calculator to compute cosecant, secant and cotangent: Unlike sin, cos and tan, most (maybe all) calculators do not have the built-in functions for csc, sec and cot. In order to evaluate, for example, csc 41, one must use sin key in the calculator by entering: 1 sin 41 (since csc 41 = 1/sin 41 ). The only time one should not use a calculator is when the angle is a special angle, i.e., 30, 45, 60 and multiples of these angles. Examples: 1. csc 41 =2. cot 65 = 3. sec 32 = 4. csc 45 = 5. cot 60 = 6. sec 30 = TrigonometrySecant, Cosecant and Cotangent (contd) Page 17

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TrigonometryIntroduction to Identities Identities When two sides of an equation are equal regardless of what the value for the variable is, its called an identity. For example, x + x = 2x is an identity. In trigonometry, we have many identities, one of which we have seen already: Reciprocal Identities Other identities includes (concluded from the right triangle above): Cofunction Identities Quotient Identities x y r 90°– Page 18

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Trigonometrysin 2 + cos 2 = 1 One Important Identity: sin 2 + cos 2 = 1 Let see if its true when = 45°, 120°, 36789°. sin 2 45° + cos 2 45° = 1sin 2 120° + cos 2 120° = 1sin ° + cos ° = 1 ??? Why is always true? x y r Write an equation that connects, y and r: Write an equation that connects, x and r: Write an equation that connects y, x and r: sin 2 + cos 2 = 1 tan = sec cot 2 = csc 2 Pythagorean Identities: Page 19

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TrigonometryProving Identities (contd) So far, weve learned some basic identities: Cofunction Identities and Reciprocal IdentitiesQuotient IdentitiesPythagorean Identities We will learn more fundamental identities, but in the mean time, lets use these ones to prove some identities that are not so obvious: 1.2. sin 2 + cos 2 = 1 tan = sec cot 2 = csc 2 Page 20

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TrigonometryProving Identities Basic Trigonometric Identities: Reciprocal IdentitiesQuotient IdentitiesPythagorean Identities sin 2 + cos 2 = 1 tan = sec cot 2 = csc 2 Techniques of proving identities: 1. Use the basic identities wisely. – Change everything in terms of sine and/or cosine – If you see (trig. function) 2, it might have to do with Pyth. Id. – Treat an identity in different ways: sin 2 + cos 2 = 1 sin 2 = 1 – cos 2 = 1 _________________ 2. Try to keep one side the same, and only simplify the complicated side or the side you can do something with. 3. If you need to change both sides, change each side independently. 4. Basic algebraic manipulation still applyFOILing, expanding, factoring, simplifying, rationalizing denominator, and combining two fractional expressions into one. 5. The goal is to show left hand side = right hand side. Ex. 3: Page 21

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TrigonometryLaw of Cosines Law of Cosines: c 2 = a 2 + b 2 – 2ab cos C To find x (of the triangle shown on the right), we use the Law of Cosines by setting up the equation as follows: x 2 = This side and this angle are opposite of each other, and one of them should be the unknown 12 30° 10 x The Law of Cosines (LoC) not only can be used to find an unknown side, it can be also used to find an unknown angle. For example, we can use LoC to find too: Why is true? Lets prove it: b a c B AC Page 22

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Trigonometry2 nd Formula for the Area of the Triangle Area of a Triangle: A = ½ab sin C (derived from A = ½bh, of course) Proof: b a c B AC h In words, the area of a triangle is _____________________________________________ Problems: ° ° Page 23

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