Trigonometry Addition Formulas and Solutions

Derivation and application of double angle formulas in trigonometry to solve various problems and equations. Understand how to find formulas for sin2A, cos2A, tan2A, rewrite trigonometric functions, and manipulate expressions. Practice solving trigonometric equations using addition formulas and demonstrate how 1 + cos⁴θ can be rewritten as 2sin²θ

• Trigonometry
• Addition Formulas
• Double Angle Formulas
• Trigonometric Equations
• Pythagorean Identities

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2. Trigonometry: Addition formulas KUS objectives BAT derive the double angle formulas BAT use the addition formulae to solve show that problems BAT use the addition formulae to solve equations Starter:

3. WB51a Trig Double Angle formulas: Find a formula for Sin2A using Sin(A + B) SinACosB + CosASinB Sin(A + A) SinACosA + CosASinA Sin2A 2SinACosA Sin2A 2SinACosA Work these out by adapting the double angle formula 1 2Sin2A SinACosA Sin4A 2Sin2ACos2A 2A 4A 2 Sin2A 2SinACosA x 3 2A = 60 3Sin2A 6SinACosA Sin60 2Sin30Cos30

4. WB51b Trig Double Angle formulas: Find a formula for Cos 2A using Cos(A + B) CosACosB - SinASinB Cos(A + A) CosACosA - SinASinA This formula comes in 3 versions Cos2A Co Cos2A Co??? ????? Replace Cos2A with (1 Sin2A) Replace Sin2A with (1 Cos2A) Cos2A (1 ???2?) ???2? Cos2A Co?2? (1 - Co?2?) Cos2A 1 2???2? Cos2A 2Co?2? 1 Work these out by adapting the double angle formula 2A 4A 2 Each has three possible versions Cos2A Co?2? ???2? x 3 2A = 60

5. WB51c Trig Double Angle formulas: Find a formula for Tan 2A using ????+???? 1 ???????? Tan (A + B) ????+???? 1 ???????? Tan (A + A) ????? ? ????? Tan 2A Tan 2A 2???30 1 ???230 1 2Tan 2A ???? 1 ???2? Tan 60 2 2A = 60 2???? 1 ???2? Tan 2A x 2 2???? 1 ???2? 2A = A 4???? 1 ???2? 2 Tan A 2Tan 2A 2

6. WB52a: Rewrite the following as a single Trigonometric function: 2sin? 2cos? 2cos? ???2? 2???????? 2 ???? 2???? 2???? 2 2???? 2???? 2???? Replace the first part = ???????? Rewrite =1 2???2?

7. Show that 1 + cos4? can be written as 2???22? WB52b: ???2? 2???2? 1 Double the angle parts ???4? 2???22? 1 1 + ???4? Replace cos4 = 1 + (2???22? 1) The 1s cancel out = 2???22?

8. WB53a: a) Given that cos? =3 4in the range [180, 360] find the exact value of 2sin2? ???? =??? ???? =??? 4 ??? ??? 7 ???? =3 7 x 4 ???? = 4 3 Use Pythagoras to find the missing side (ignore negatives) ???2? Cosx is positive so in the range 270 - 360 y = Cos 7 Therefore, Sinx is negative ???? = 90 180 270 360 4 2 Sin2x 4SinxCosx y = Sin Sub in Sinx and Cosx 2 Sin2x = 4 3 7 4 4 Work out and leave in surd form 2 Sin2x = 3 7 4

9. WB53b: b) Given that cos? =3 4in the range [180, 360] find the exact value of tan2? ???? =??? ???? =??? 4 ??? ??? 7 ???? =3 7 x 4 ???? = 3 3 Use Pythagoras to find the missing side (ignore negatives) Cosx is positive so in the range 270 - 360 y = Cos 7 Therefore, Tanx is negative ???? = 90 180 270 360 3 2???? 1 ???2? Tan 2x Sub in Tanx y = Tan 7 90 180 270 360 2 3 Tan 2x = 7 7 1 3 Work out and leave in surd form 3 ???2? = 3 7

10. 2 Show that tan2? = cot ? tan ? WB54: 2???? 1 ???2? ???2? Divide each part by tan 2???? ???? ???2? ???? ???2? 1 Rewrite each part Cancel terms ???? 2 ???2? ???? ????

11. WB55: Given that x = 3sin? and ? = 3 4cos2? Eliminate and express y in terms of x ???2? = 1 2???2? ? = 3???? Divide by 3 ? 3= ???? Replace Cos2 and Sin 2 3 ? 4 ? 3 = 1 2 Multiply by 4 ? = 3 4???2? Subtract 3, divide by 4 Multiply by -1 2 ? 3 3 ? 4 3 ? = 4 8 = ???2? Subtract 3 2 ? 3 ? = 1 8 Multiply by -1 2 ? 3 ? = 8 1

12. WB56a: Solve the equation 3cos2? ???? + 2 = 0 in the range 0 ? 360 3(2???2? 1) ???? + 2 = 0 ???2? 2???2? 1 6???2? ???? 1 = 0 (3???? + 1)(2???? 1) = 0 y = Cos 12 13 ???? = 1 3 or ???? =1 2 90 180 270 360 ? = 60 ,109.5 ,250.5 ,300

13. WB56b: Solve the equation cos2x = 7cosx + 3 for 0 ? 2 2???2? 1 = 7cos? + 3 ???2? 2???2? 1 2???2? 7cos? 4 = 0 (2cos? + 1)(cos? 4) = 0 cos? = 1 2,4 x=2? 3,4? 3

14. WB56c: Solve the equation sin2x = tanx for 0 ? 2 2sin?cos? =sin? ???2? 2sin???? ? cos? 2sin????2? = sin? sin?(2???2? 1) = 0 sin? = 0 or ???2? =1 2 2 sin? = 0 or cos? = 2 x=? 4,3? 4,5? 4,7? x= 0,?,2? 4

15. KUS objectives BAT derive the double angle formulas 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 ____