# Further Trigonometric Identities and their Applications Introduction This Further Trigonometric Identities and their Applications Introduction This chapter extends your knowledge of Trigonometrical identities You will see how to solve equations involving combinations of sin, cos and tan You will learn to express combinations of these as a transformation of a single graph Teachings for Exercise 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Q By GCSE Trigonometry: 1 B So the coordinates of P are: P 1 A O M N So the coordinates of Q are: Q P 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae

2= 2=( 2 2+ 2 )+( 2 2+ 2 ) Multiply out the brackets Rearrange 2=( 2 +2 )+( 2 + 2 )2(+) 1 2=22(+) 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Q You can also work out PQ using the triangle OPQ: Q 1 B 1 B-A P P 1 A O 1 M N 2bcCosA 2Cos(B - A) Sub in the values

Group terms 2Cos(B - A) 2Cos(A - B) Cos (B A) = Cos (A B) eg) Cos(60) = Cos(-60) 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae 2=22(+) 22(+) 2(+) + 2Cos(A - B) 2Cos(A - B) 2Cos(A - B) Cos(A - B) Subtract 2 from both sides Divide by -2 Cos(A - B) = CosACosB + SinASinB Cos(A + B) = CosACosB - SinASinB 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Cos(A - B) CosACosB + SinASinB Cos(A + B) CosACosB - SinASinB Sin(A + B) SinACosB + CosASinB Sin(A - B) SinACosB - CosASinB 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition

formulae Sin(A + B) = SinACosB + CosASinB Tan (A+B) Rewrite Tan (A+B) Sin(A - B) = SinACosB - CosASinB Cos(A - B) = CosACosB + SinASinB Cos(A + B) = CosACosB - SinASinB Show that: Tan (A + B) Tan (A+B) Tan (A+B) TanA+ TanB 1 Divide top and bottom by CosACosB Simplify each Fraction TanATanB Tan 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Cos(A - B) CosACosB + SinASinB Cos(A + B) CosACosB - SinASinB Sin(A + B) SinACosB + CosASinB Sin(A - B) SinACosB - CosASinB Tan (A + B) Tan (A - B) You may be asked to prove either of the Tan identities using the Sin

and Cos ones! 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Cos(A + B) CosACosB - SinASinB 15=(45 30) Sin(A - B) SinACosB - CosASinB Cos(A - B) CosACosB + SinASinB Sin(A + B) SinACosB + CosASinB Sin(45 - 30) Sin45Cos30 Cos45Sin30 Sin(A - B) SinACosB - CosASinB Sin(45 - 30) Tan (A + B) Tan (A - B) Sin(45 - 30) Show, using the formula for Sin(A B), that: 6 2 15= Sin(15) 2 3 2 1 2 2 2 2 6 2 4 4 6 2 A=45, B=30 These can be written as surds Multiply each pair Group the

fractions up 4 4 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae = = that: < AGiven < 270 3 5 = 5 3 A = 3 4 4 Cos Find the value of: Tan(A+B) Use Pythagoras to find the missing side (ignore negatives) Tan is positive in the range 180 270

= Tan (A + B) = 90 18 0 270 y= 360 Tan 12 13 13 = 5 = B 12 5 12 = 5 12 Use Pythagoras to find the missing side (ignore negatives) Tan is negative in the range 90 180 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae

Tan (A + B) Given that: < A < 270 Tan (A + B) Cos Find the value of: Tan (A + B) Substitute in TanA and TanB Work out the Numerator and Denominator Tan(A+B) Tan (A + B) 3 = 4 5 = 12 Leave, Change and Flip Tan (A + B) Simplify Tan (A + B) Although you could just type the whole thing into your calculator, you still need to show the stages for the workings marks 7A Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Given that: 2 ( + )=3 ( ) Express Tanx in terms of Tany 2 ( + )=3 ( )

2(+)3 (+) 2+23 +3 2+23 +3 2+23 +3 23 3 2 (23 ) 3 2 3 2 23 Rewrite the sin and cos parts Multiply out the brackets Divide all by cosxcosy Simplify Subtract 3tanxtany Subtract 2tany Factorise the left side Divide by (2 3tany) 7A Teachings for Exercise 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Sin(A + B) SinACosB + CosASinB Replace B with A Sin(A + A) SinACosA + CosASinA Simplify Sin2A 2SinACosA Sin2A SinACosA 2 Sin4A 2Sin2ACos2A 2A 4AA Sin2A 2SinACosA x3 3Sin2A 6SinACosA 2A = 60

Sin60 2Sin30Cos30 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Cos(A + B) CosACosB - SinASinB Replace B with A Cos(A + A) CosACosA - SinASinA Simplify Cos2A Co Cos2A Co Replace Cos2A with (1 Sin2A) Cos2A (1 Cos2A 1 Replace Sin2A with (1 Cos2A) Cos2A Co1 - Co Cos2A 2Co 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Tan (A + B) Replace B with A Tan (A + A) Simplify Tan 2A Tan 2A 2 2A = 60 Tan 60 Tan 2A

x2 2Tan 2A 2A = A Tan A 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Rewrite the following as a single Trigonometric function: 2 2 2 2 2 2 2 2 2 2 2 2 Replace the first part Rewrite 1 2 2 7B Further Trigonometric Identities and their

Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae 1+ 4 Show that: Can be written as: 2 2 2 2 2 2 1 4 2 2 2 1 1+ 4 1+(2 2 2 1) 2 2 2 Double the angle parts Replace cos4 The 1s cancel out 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae = = 3 4 7 180 <<360

Find the exact value of: 2 = x Given that: 3 = 4 = 4 3 7 4 Use Pythagoras to find the missing side (ignore negatives) Cosx is positive so in the range 270 360 7 Therefore, Sinx is negative = 4 90 18 0 270 y= 360 Cos Sin2x 2SinxCosx Sin2x = 2 y= Sin Sin2x = Sub in Sinx and Cosx

Work out and leave in surd form 7B Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae = = 3 4 Given that: 3 = 4 180 <<360 Find the exact value of: 2 = 4 7 x 3 7 = 3 Use Pythagoras to find the missing side (ignore negatives) Cosx is positive so in the range 270 360 7 Therefore, Tanx is negative =

3 90 90 18 0 18 0 270 y= 360 Cos Tan 2x Tan 2x 270 y= 360 Tan 2 =3 7 Sub in Tanx Work out and leave in surd form 7B Teachings for Exercise 7C Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Prove the identity: 2 2 2 2 1 2

2 2 2 1 2 Divide each part by tan Rewrite each part 2 7C Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities By expanding: ( 2 + ) Show that: ( 3 ) 3 4 3 ( + ) + Replace A and B ( 2 + ) 2 + 2 ( 3 ) (2)+(1 2 2 ) ( 3 ) 2 2 + 2 3 ( 3 ) 2 (1 2 )+ 2 3 Replace Sin2A and Cos 2A Multiply out Replace cos2A Multiply out ( 3 ) 22 3 + 2 3

( 3 ) 3 4 3 Group like terms 7C Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Given that: =3 and =3 4 2 Eliminate and express y in terms of x =3 = 3 Divide by 3 =3 4 2 3 =2 4 2 =1 2 2 3 =1 2 3 4 2 ( ) Replace Cos2 and Sin Multiply by 4 3 =4 8 3 ( )

2 2 Subtract 3 =1 8 ( 3 ) 2 = 8 ( 3 ) 1 Multiply by 1 Subtract 3, divide by 4 Multiply by -1 7C Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Solve the following equation in the range stated: 3 2 +2=0 0 360 3 2 +2=0 3(2 2 1) +2=0 6 2 3 +2=0 1 2 1 3 90 18 0 270 360

Multiply out the bracket Group terms 2 6 1=0 Factorise (3 +1)(2 1)=0 = (All trigonometrical parts must be in terms x, rather than 2x) y= Cos Replace cos2x 1 3 or = 1 2 Solve both pairs 1 = 1 3 ( ) 1 = 1 2 () =109.5 , 250.5 =60 , 300 Remember to find

additional answers! =60 ,109.5 ,250.5 ,300 7C Teachings for Exercise 7D Further Trigonometric Identities and their Applications You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Show that: 3 +4 Can be expressed in the form: ( +) ( +) + 3 +4 + =3 = =4 3 4 = ( ) ( ) So in the triangle, the Hypotenuse is R Replace with the expression Compare each term they must be equal! 4 3

= 32 +42 =5 3 3 = 5 = So: 3 +4 5 sin (+53.1 ) 3 = 5 R=5 Inverse Cos 1 =53.1 Find the smallest value in the acceptable range given 7D Further Trigonometric Identities and their Applications You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Show that can express: you 3 ( ) 3 = 3 =1

2 = 1 + ( 3 ) 2 Replace with the expression Compare each term they must be equal! =2 In the form: ( ) =1 2 =1 = So: 3 ( 2 sin 3 ) 3 Divide by 2 1 2 = = R=2 1 1 2 Inverse

cos Find the smallest value in the acceptable range 7D Further Trigonometric Identities and their Applications You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Sketch the graph of: 3 ( = Sketch the graph of: 2 sin ) y=sin 1 Show that can express: you 3 3 -1 /2 3 /2 Start out with sinx

2 In the form: ( ) ( y =sin 1 So: 3 ( 2 sin 3 ) /3 /2 4 /3 3 /2 2 ( ) At the yintercept, x=0 -1 -2 )

3 ) 2 ( y =2 sin 1 2 sin 3 3 -1 3 /3 /2 4 /3 3 /2 2 Translate /3 units right Vertical stretch, scale factor 2 7D Further Trigonometric Identities and their

Applications You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Express: 2 +5 ( ) + 2+5 + =2 = 22 +52 =5 Replace with the expression Compare each term they must be equal! = 29 in the form: ( ) So: 2 +5 29 cos( 68.2) =2 29 =2 = = 2 29 1 =68.2 2 29 R= 29 Divide by 29 Inverse cos

Find the smallest value in the acceptable range 7D Further Trigonometric Identities and their Applications You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Solve in the given range, the following equation: 2 +5 =3 0 <<360 29 cos( 68.2)=3 Divide by 29 cos ( 68.2)= 3 29 1 68.2= Inverse Cos 3 29 68.2=56.1, 56.1, 303.9 We just showed that the original equation can be rewritten 2 +5 = 29cos( 68.2) Remember to work out other values in the adjusted range Add 68.2 (and put in order!) =12.1, 124.3 Hence, we can solve this equation instead!

29 cos( 68.2)=3 0 <<360 68.2 < 68.2<291.2 56.1 Remember to adjust the range for ( 68.2) -90 56. 1 90 303. 9 18 0 270 y= Cos 360 7D Further Trigonometric Identities and their Applications Rcos( ) chosen as ) chosen as it gives us the same form as the expression You can write expressions of the form acos + bsin, where a and b are constants, as a sine or cosine function only Find the maximum value of the following expression, and the smallest positive value of at which it arises: 12 +5 13 cos ( 22.6) 13 cos ( 22.6)

13(1) =13 22.6=0 =22.6 Max value of cos( - 22.6) = 1 Overall maximum therefore = 13 Cos peaks at 0 = 22.6 gives us 0 ( ) + 12 +5 + =12 =5 Replace with the expression Compare each term they must be equal! = 122 +52 =13 =12 13 =12 12 = 13 = 1 =22.6 12 13 R = 13 Divide by 13 Inverse cos Find the smallest value in the acceptable range 7D Further Trigonometric Identities and their Applications You can write expressions of

the form acos + bsin, where a and b are constants, as a sine or cosine function only ( ) ( ) Whichever ratio is at the start, change the expression into a function of that (This makes solving problems easier) Remember to get the + or signs the correct way round! 7D Teachings for Exercise 7E Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae +=2 ( =2 + 2 2 + 2 2 ) ( ) +=2

+ 2 2 ) = 2 ( ) ( ( ) ( ) + 2 2 ( ) ( ) You get given all these in the formula booklet! 7E Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae +=2 ( =2 + 2 2

) ( ) + 2 2 ) ( ) ( 1) 2) ( )( ( + 2 2 ) ( ( + ) =+ ( ) = ) Add both sides together (1 + 2) Let (A+B) = P Let (A-B) = Q 1) 2) ( + ) + ( )=2 + + 2 2 2 +=2

( + ) + ( )=2 + +=2 2 2 ( ) ( ) + = 2 ( 2 ) ( 2 ) Using the formulae for Sin(A + B) and Sin (A B), derive the result that: ) + = = 2 = + = + 2 + = 2) = 2 = 1) 1+ 2 Divid e by 2 = 2 1-2 Divid e by 2 7E Further Trigonometric Identities and their Applications Show that:

You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae +=2 ( ( + ) 2 2 ) =2 ( + 2 2 ) ( ) + 2 2 ( ) ( ) + = 2 ( 2 ) ( 2 ) +=2 =2 105 15= ( + 2 2

) ( 1 2 ) P = 105 105 +15 105 15 105 15=2 2 2 ( ) ( 105 15=260 45 105 15=2 1 1 2 1 2 ) Q = 15 Work out the fraction parts Sub in values for Cos60 and Sin45 Work out the right hand side 105 15= 2 7E Further Trigonometric Identities and their Applications Solve in the range indicated: You can express sums and 4 3 =0 0

differences of sines and cosines as products of sines and cosines by + =2 using the factor formulae ( ) ( 2 ) 4 +3 4 3 4 3 =2 ( 2 ) ( 2 ) 7 4 3 =2 ( ) ( ) 2 2 7 2 ( ) ( )=0 2 2 7 ( =0 2 ) + +=2 2 2 ( ) ( ) + =2 ( 2 ) ( 2 ) + +=2 ( ( 2 ) 2 ) + = 2 (

2 ) ( 2 ) 0 0 /2 3 /2 Q = 3 Work out the fractions Either the cos or sin part must equal 0 7 = 1 0 2 2 P = 4 Set equal to 0 Adjust the range 7 7 0 2 2 2 y= Cos 7 3 5 7 = , , , 2 2 2 2 2 3 5 , = ,

, 7 7 7 Inverse cos Solve, remembering to take into account the different range Once you have all the values from 0-2, add 2 to them to obtain equivalents Multiply by 2 and divide by 7 7E Further Trigonometric Identities and their Applications Solve in the range indicated: You can express sums and 4 3 =0 0 differences of sines and cosines as products of sines and cosines by + =2 using the factor formulae ( ) ( 2 ) 4 +3 4 3 4 3 =2 ( 2 ) ( 2 ) 7 4 3 =2 ( ) ( ) 2 2 7 2 ( ) ( )=0 2 2 ( ) =0 2 + +=2 2

2 ( ) ( ) + =2 ( 2 ) ( 2 ) + +=2 ( ( 2 ) 2 ) + = 2 ( 2 ) ( 2 ) 0 0 /2 3 /2 Q = 3 Work out the fractions Either the cos or sin part must equal 0 = 1 0 2 2 P = 4 Set equal to 0 Adjust the range

0 2 2 2 y= Sin =0 2 =0 Inverse sin Solve, remembering to take into account the different range Once you have all the values from 0-2, add 2 to them to obtain equivalents Multiply by 2 7E Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae + 2 2 ( ) ( ) + =2 ( 2 ) ( 2 ) + +=2 ( ( 2 ) 2 ) + = 2 (

2 ) ( 2 ) +=2 Prove that: ( +2 )+ ( + ) + = ( + ) ( +2 ) + ( + ) + Numerator: In the numerator: ( +2 )+ ( + ) + ( +2 )+ Ignore sin(x + y) for now + +=2 2 2 ( ) ( ( +2 + +2 2 2 ( +2 )+=2 2 ( + ) 2 ( + ) + (+ ) ( + ) (2 +1) ) Use the identity for adding 2 sines )( ) P = x + 2y Q=x

Simplify Fractions Bring back the sin(x + y) we ignored earlier Factorise ( + ) (2 +1) 7E Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae + 2 2 ( ) ( ) + =2 ( 2 ) ( 2 ) + +=2 ( ( 2 ) 2 ) + = 2 ( 2 ) ( 2 ) +=2 Prove that: ( +2 )+ ( + ) + = ( + ) ( +2 ) + ( + ) + Numerator: In the denominator: ( +2 )+ ( + )+ ( +2 )+

Ignore cos(x + y) for now + +=2 2 2 Use the identity for adding 2 cosines ( ) ( ) +2 + +2 ( +2 )+=2 ( 2 )( 2 ) 2 ( + ) 2 ( + ) + (+ ) ( + ) (2 +1) P = x + 2y Q=x Simplify Fractions Bring back the cos(x + y) we ignored earlier Factorise ( + ) (2 +1) Denominator ( + ) (2 +1) : 7E Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the factor formulae + 2 2 ( ) ( ) +

=2 ( 2 ) ( 2 ) + +=2 ( ( 2 ) 2 ) + = 2 ( 2 ) ( 2 ) +=2 ( +2 )+ ( + ) + ( +2 ) + ( + ) + ( + )(2 +1) (+ )(2 +1) ( + ) ( + ) ( + ) Replace the numerator and denominator Cancel out the (2cosy + 1) brackets Use one of the identities from C2 Prove that: ( +2 )+ ( + ) + = ( + ) ( +2 ) + ( + ) + Numerator: ( + ) (2 +1) Denominator ( + ) (2 +1) :

7E Summary We have extended the range of techniques we have for solving trigonometrical equations We have seen how to combine functions involving sine and cosine into a single transformation of sine or cosine We have learnt several new identities