Trigonometry Addition Formulas: Proofs and Applications
Explore trigonometry addition formulas through a series of proofs and practical applications, showcasing how to use these formulas to solve equations and problems effectively. Includes a detailed breakdown of identities and their derivations for comprehensive learning.
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sin ? ? =sin?cos? sin?cos? cos ? ? =cos?cos? sin?sin? tan ? ??? ? 1 tan ? tan ? tan ? ? = Proofs
Trigonometry: Addition formulas KUS objectives BAT use the addition formulae to solve show that problems BAT use the addition formulae to solve equations Starter: True or false? tan ? = tan(? ??) tan ? 2 = +tan(5? 2) FALSE tan ? 2 = tan(5? 2) cos ? = cos( ?) cos 8 3? = cos(3? 8) TRUE cos 8 3? = cos(3? 8) 5? 17 22? 17 sin = sin sin ? = sin(? + ?) 5? 17 TRUE 22? 17 sin = sin Check with a calculator
WB46a: Show that 6 2 6 2 ?) co?75 = ?) sin15 = 4 4 ?) ???15 = ???(45 30) Sin(A - B) SinACosB - CosASinB Sin(45 - 30) = Sin45Cos30 Cos45Sin30 2 3 2 1 2 ??? 45 30 = 2 2 2 6 2 6 2 4 = 4 4= QED
WB41b: Show that 6 2 6 2 ?) co?75 = ?) sin15 = 4 4 b) cos75 = ???(45 + 30) ??? 45 + 30 = cos45cos30 sin45sin30 2 3 2 1 2 ??? 45 + 30 = 2 2 2 6 2 6 2 4 = 4 4= QED
WB47: a) Use the identity sin(? + ?) = sin? cos? + sin? cos? to show that 2sin ? +? 3 = sin? + 3cos? b) Use the identity ???(? ?) = ???? cos? + sin? sin? to show that sin? 2??? ? 5? = 3cos? 6 ?) sin ? +? = sin? cos? 3+ sin? 1 2 3 3cos? = sin? + cos? 3 2 So 2 sin ? +? = sin? + 3cos? 3 3 1 2 ?) cos ? ? = cos? cos5? 6+ sin? cos5? = cos? + sin? 2 6 6 So sin x 2 cos ? ? = sin? 3cos? + sin? = 3cos? 6
WB48: Given that sin? = 3 where B is Obtuse . Find the value of tan ? + ? and cos? = 12 5 in the range 180 < ? < 270 13 ???? =??? ???? =??? ??? ??? 5 3 ???? =3 ???? = 3 4 5 A 4 Use Pythagoras to find the missing side (ignore negatives) y = Tan 9 0 18 0 270 360 Tan is positive in the range 180 - 270 ???? =??? ??? 13 ???? =??? 5 ??? 5 12 B ???? = ???? = 12 12 13 ???? = 5 12 y = Tan Use Pythagoras to find the missing side (ignore negatives) 9 0 18 0 270 360 Tan is negative in the range 90 - 180 ????+???? 1 ???????? Tan (A + B)
WB48 (cont): Given that sin? = 3 where B is Obtuse . Find the value of tan ? + ? and cos? = 12 5 in the range 180 < ? < 270 13 ????+???? 1 ???????? Tan (A + B) Substitute in TanA and TanB 3 4+ 5 1 12 Tan (A + B) 3 4 5 Work out the Numerator and Denominator 12 1 3 63 48 Tan (A + B) Leave, Change and Flip Tan (A + B) 1 3 48 63 Simplify Tan (A + B) 16 63 Although you could just type the whole thing into your calculator, you still need to show the stages for the workings marks
WB49: Given that 2 sin ? + ? = 3cos ? ? Express tan? in terms of tan? 2??? ? + ? = 3???(? ?) Rewrite the sin and cos parts 2(???????? + ????????) = 3(???????? + ????????) Multiply out the brackets 2???????? + 2???????? = 3???????? + 3???????? Divide all by cosxcosy 2???????? + 2???????? = 3???????? + 3???????? ???????? ???????? ???????? ???????? Simplify 2???? + 2???? = 3+3???????? Subtract 3tanxtany Subtract 2tany 2???? 3???????? = 3 2???? Factorise the left side ????(2 3????) = 3 2???? Divide by (2 3tany) ???? = 3 2???? 2 3????
WB50a: Solve each of the following equations for in the interval 0 2 tan 3?+tan? 1 tan 3? tan? =3 a) ??? ? ? 4 = ???? b) 2 3 4 ?) sin ? ? = sin? cos? 3 sin? 3cos? =1 3 2sin? 2cos? 3 So 1 3 2sin? 2cos? = cos x Rearrange to 1 3 2sin? = 1 + cos x 2 3 Divide through by cos x tan? = 2 1 + = 2 + 3 2 Solve to give ? =5? 17? 12 12,
WB50b: Solve each of the following equations for in the interval 0 2 tan 3?+tan? 1 tan 3? tan? =3 a) ??? ? ? 4 = ???? b) 2 3 4 b) ??? = tan 3? +? 4 So tan 3? +? =3 4 2 Solve to give 3? +? 4 = 0.983, 4.124 ,7.266, 10.408, Solve to ? = 0.066, 1.113, 2.160,
KUS objectives BAT use the addition formulae to solve show that problems BAT use the addition formulae to solve equations self-assess using: R / A / G I am now able to ____ . To improve I need to be able to ____
