Ch:7 Trigonometric Identities and Equations

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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 Sin ( +θ) = - sin θ Tan ( +θ) = tan θ Cos ( θ)= cos θ Sin( θ)= -sin θ tan( θ)= -tan θ

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

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

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

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

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)

Question Time!!!

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

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

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

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

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

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

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

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

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

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

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