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Ch:7 Trigonometric Identities and Equations

Ch:7 Trigonometric Identities and Equations . By: Linitha and Hina . 7.1 Exploring Equivalent Trigonometric Functions . Related functions with and 2 Cos ( – θ )= - cos θ Sin ( – θ ) = sin θ Tan ( – θ ) = - tan θ Cos ( + θ ) = - cos θ

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Ch:7 Trigonometric Identities and Equations

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  1. Ch:7 Trigonometric Identities and Equations By: Linitha and Hina

  2. 7.1 Exploring Equivalent Trigonometric Functions • Related functions with and 2 • Cos ( – θ)= - cos θ • Sin ( – θ) = sin θ • Tan ( – θ) = - tan θ • Cos ( + θ) = - cos θ • Sin ( +θ) = - sin θ • Tan ( +θ) = tan θ • Cos (2 + θ)= cos θ • Sin(2 + θ)= -sin θ • tan(2 + θ)= -tan θ

  3. 7.2 Compound Angle Formulas • Addition formulas • Sin (a+b) = sin a cos a + cos a sin b • Cos (a+b) = cos a cos b – sin a sinb • Tan (a+b) = tan a +tan b / 1- tan a tan b • Subtraction formulas • Sin (a-b)= sin a cos b – cos a sin b • Cos (a-b) = cos a cos b +sin a sin b • Tan (a-b) = tan a – tan b/ 1 + tan a tan b

  4. 7.3 Double Angle Formulas • Double angle formula for sine • Sin 2θ = 2 sin θ cos θ • Double angle formulas for cosine • Cos 2θ = cos2 θ – sin2θ • Cos 2θ = 2 cos2 θ – 1 • Cos 2θ = 1-2 sin2 θ • Double angle formulas for tangent • Tan 2θ = 2 tan θ / 1- tan2 θ

  5. 7.4 Proving Trigonometric Identities Reciprocal identities • Csc x= 1/ sin x • Sec x= 1/cos x • Cot x = 1/tan x Quotient identities • Tan x = sinx / cos x • Cot x= cos x/ sinx Pythagorean identities • Sin 2 x + cos 2 x = 1 • 1 + tan 2 x = sec 2 x • 1+ cot x = csc 2 x Double angle formulas • Sin 2x = 2 sinx cosx • Cos 2x = cos2x– sin2x • Cos 2x = 2 cos2x – 1 • Cos 2x = 1-2 sin2x • Tan2x = 2 tan x/ 1- tan2x Addition /subtraction formulas • Sin (x+y) = sin x cos y + cos x sin y • Cos (x+y) = cos x cos y – sin x sin y • Tan (x+y) = tan x +tan y / 1- tan x tan y • Subtraction formulas • Sin (x-y)= sin x cos y – cos x sin y • Cos (x-y) = cos x cos y +sin x sin y • Tan (x-y) = tan x – tan y/ 1 + tan x tan y

  6. 7.5 Solving Linear Trigonometric Equations • Special Triangles • CAST Rule • Calculator (only when not in special triangle) • Period of the function so the number of solutions are known in the specified interval

  7. 7.6 Solving Quadratic Trigonometric Equations • Factoring • Quadratic Formula Sin2 x – sinx = 2 Sin2 x – sinx – 2 = 0 ( sinx – 2) (sinx + 1) = 0 Sinx = 2 or sinx = -1 No solution x = 3 2 (0, -1)

  8. Question Time!!!

  9. 1. Use the co function identities to write an expression that is equivalent to each of the following expressions. • Sin 6 • Tan 3 8 • Cos 5 18

  10. 2. State whether each of the following are true or false • Cos (θ +2 )= cos θ • Sin ( - θ) = -sin θ • Cot ( + θ)= tan θ 2

  11. 3. Determine the exact value of • A) Cos (15 °) B) tan(-5 /12) • 4. simplify each expression • A) cos 7 /12 cos 5 /12 + sin 7 /12 sin 5 /12 • B) sin 2x cos x – cos 2x sin x

  12. 5. Simplify each of the following expressions and then evaluate • A) 2 sin /8 cos /8 • B) 2 tan /6 / 1 – tan 2 /6

  13. 6. If cosθ = -2/3 and 0 < θ < 2pie , determine the value of cos 2θ and sin 2θ • 7. Develop a formula for sin x/2

  14. 8. prove that sin 2x / 1 + cos2x = tan x • 9. prove that sin x + sin 2x = sin 3x is not an idenitity • 10. prove that cos ( /2 + x) = - sin x

  15. 11. Cos (x - y)/ cos (x + y) = 1 + tan x tan y/ 1- tan x tan y • 12.Prove that tan 2x – 2 tan 2x sin2 x = sin 2x • 13. prove that 1 + tan x / 1 + cot x = 1- tan x /cot x - 1

  16. 14. Determine all solutions in the specified interval for the following equation: 0 < x < 2 • 2sinx + 1 = 0

  17. 15. Use a calculator to determine the solutions for the following equation on the interval 0 < x < 2 • 2 – 2cotx = 0

  18. 16. Solve the equation for x in the interval 0 < x < 2 • 2sin2x – 3sinx + 1 = 0

  19. 17. Use a trigonometric identity to create a quadratic equation. Then solve the equation for x in the interval [0, 2 ] • 2sec2x – 3 + tanx = 0

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