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**Ch:7 Trigonometric Identities and Equations**

By: Linitha and Hina

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**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 ( θ)= cos θ Sin( θ)= -sin θ tan( θ)= -tan θ

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**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

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**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 – sin2 θ 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

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**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 sin x cos x Cos 2x = cos2x– sin2 x Cos 2x = 2 cos2 x – 1 Cos 2x = 1-2 sin2 x 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

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**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

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**7.6 Solving Quadratic Trigonometric Equations**

Factoring Quadratic Formula Sin2 x – sinx = 2 Sin2 x – sinx – 2 = 0 ( sinx – 2) (sinx + 1) = 0 Sinx = or sinx = -1 No solution x = 3 2 (0, -1)

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Question Time!!!

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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

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**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

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**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

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**5. Simplify each of the following expressions and then evaluate**

A) 2 sin /8 cos /8 B) 2 tan /6 / 1 – tan 2 /6

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**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

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**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

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**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

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**14. Determine all solutions in the specified interval for the following equation:**

0 < x < 2 2sinx + 1 = 0

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

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**16. Solve the equation for x in the interval 0 < x < 2**

2sin2x – 3sinx + 1 = 0

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**17. Use a trigonometric identity to create a quadratic equation**

17. Use a trigonometric identity to create a quadratic equation. Then solve the equation for x in the interval [0, 2 ] 2sec2 x – 3 + tanx = 0

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