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Comprehensive Guide to Trigonometric Identities and Equations

This resource explores fundamental trigonometric identities and equations, including equivalent functions, co-function identities, and compound angle formulas. Key topics covered include addition and subtraction formulas, double angle formulas, and methods for proving identities. Additionally, it delves into solving linear and quadratic trigonometric equations using various techniques such as factoring and the quadratic formula. Ideal for students seeking to deepen their understanding of trigonometry and enhance their problem-solving skills.

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Comprehensive Guide to 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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