Precal Chapter 5 Trig Identities and Product-to-Sum Identities

x can only be between -1 and 1 in ___

"Find all solutions of each equation in the interval [0,2π)" questions

Pythagorean identity cot and csc

1+cot²θ=csc²θ

Pythagorean identity tan and sec

1+tan²θ=sec²θ

5) Prove the Identity; (sinx/1-cosx)+(1-cosx/sinx)=2cscx

1. Common denominator; (sinx⋅sinx/sinx(1-cosx))+((1-cosx)(1-cosx)/sinx(1-cosx)) 2. Combine; (sin²x+1-cosx-cosx+cos²x/sinx(1-cosx) 3. Simplify; (2-2cosx/sinx(1-cosx) 4. Common Factor; (2(1-cosx)/sinx(1-cosx)) 5. Simplify; 2/sinx 6. Reciprocal Identity; 2cscx

13) Use a sum/difference identity to find sin(π/12)

1. Decide what to add or subtract to get π/12. (In this case subtract because all the unit circle fractions are bigger than π/12; π/12=(π/4)-(π/6) 2. Subtraction, so use sin(α-β)=sinαcosβ-cosαsinβ; sin((π/4)-(π/6))=sin(π/4)cos(π/6)-cos(π/4)sin(π/6) =(√2/2)⋅(√3/2)-(√2/2)(1/2) =(√6/4)-(√2/4)=((√6-√2)/4)

17) Solve on the interval [0,2π); sin2x=-cosx

1. Double angle Identity sin2θ=2sinθcosθ; 2sinxcosx=-cosx 2. solve; cosx(2sinx-1)=0 3. Set equal to 0 cosx=0 2sinx+1 x is 0 at (π/2), (3π2) sinx=(-1/2) x=(5π/6), (7π/6)

10) Find all solutions of each equation in the interval [0,2π); cos²x+cosx-2=0

1. Factor; (cosx+2)(cosx-1)=0 2. Set both equal to zero cosx+2=0 cosx-1=0 cosx=-2 cosx=1 not between -1 and 1 x is only 1 at 0° x=0

1) Use a Pythagorean identity to find tanθ if secθ=-5 and sinθ>0

1. Find pythagorean identity that matches; 1+tan²θ=sec²θ 2. solve tanθ=±√24 3. determine positive or negative (-) (sinθ>0 and secθ is negative) 4. SIMPLIFY RADICAL tanθ=-2√6

2) Simplify secx-sinxtanx

1. Is there anything algebraic you can do? No 2.Change into sin's and cos's. (1/cosx)-sinx⋅(sinx/cosx) 3. Simplify (1/cosx)-(sin²x/cosx) 4. Combine (1-sin²x/cosx) 5. pythagorean identity (cos²x/cosx) 6. simplify; cosx

3) Simplify (sin²x/sec²x-1)

1. Pythagorean Identity (sin²x/tan²x) 2. Change into sins and cos's. sin²x÷(sin²x/cos²x) 3. Simplify cos²x

4) Prove the Identity; csc²x(1-sin²x)=cot²x

1. Pythagorean Identity; csc²x⋅cos²x 2. Reciprocal Identity; (1/sin²x)⋅cos²x 3. combine (cos²x/sin²x) 4. quotient identity; cot²x

8) Solve for all values of x; 2=4sin²x-1

1. Solve for sin; sinx=±√3/4 2. sin=y, y is √3/4 at the 60° angles x= (π/3)+2πn, (2π/3)+2πn, (4π/n)+2πn, (5π/n)+2πn

11) Find all solutions of each equations in the interval [0,2π); 2cos²x=sinx-1

1. There are two different trig functions, and that only works when you have a common factor that you can factor out. So instead we have to make a substitution to get rid of one; 2(1-sin²x)=sinx+1 2. Distribute; 2-2sin²x=sinx-1 3. All units to one side; 0=2sin²x+sinx-1 5. Factor (sinx+1)(2sinx-1) 4.Set both equal to 0 sinx-1=0 2sinx-1=0 sinx=-1 sinx=-1/2 y=-1 at 3π/4 y=-1/2 at π/6 and 5π/6 x= (3π/2), (π/6), (5π/6)

14) Find the exact value of cos(7π/12)cos(π/4)+sin(7π/12)sin(π/4)

1. Use cosαcosβ+sinαsinβ=cos(α-β) and solve for cos(α-β) cos((7π/12)-(π/4))=cos((7π/12)-(3π/12)=cos(4π/12)=cos(π/3) 2. at π/3, x=1/2 x=1/2

15) Simplify (tan9x+tan2x)/(1-tan9xtan²x)

1. Use tan(α+β)=(tanα+tanβ)/(1-tanαtanβ) 2. 9x=α, and tx=β 3. tan(9x+2x)=tan(11x)

18) Use a half angle identity to find the exact value of sin112.5°

1. Use the sin equation; sinθ/2=±√(1-cosθ/2) 2. 112.5° is in quadrant 2, sin is positive 3. Solve for θ, 112.5=(θ/2) 225°=θ 4. sin 112.5°=√(1-cos225°/2) =(√(1-(-√2/2))/2 ×2 =√((2+√2)/4) =√(2+√2)/2

16) Find the value of sin2θ, cos2θ, and tan2θ for the given value and interval; cosθ=(3/5) (270°, 360°)

1. draw a reference triangle 2. Double angle identity sin2θ=2sinθcosθ; =2(-4/5)(3/5) =(-24/25) 3. Double angle identity cos2θ=cos²θ-sin²θ =(3/5)²-(-4/5)² =(9/25)-(16/25) =(-7/25) 4. Use tan2θ=(sin2θ)/(cos2θ) =(-24/25)÷(-7/25) =(-24/-7) =(24/7)

6) Prove the Identity; sec⁴x-tan⁴x=sec²x+tan²x

1. factor (sec²x+tan²x)(sec²x-tan²x) 2. Pythagorean Identity; (sec²x+tan²x)(1) 3.Simplify; sec²x+tan²x

7) Prove the Identity; tanxcsc²x-tanx=cotx

1. factor; tanx(csc²x-1) 2. Pythagorean Identity; tanx⋅cot²x 3. tanx⋅cotx⋅cotx 4. Reciprocals; 1⋅cotx 5. cotx

12) Find all solutions of each equation in the interval [0,2π); cscx-cotx=1

1. move cot to other side; cscx=1+cotx 2. square both sides; csc²x=1+2cotx+cot²x 3. Pythagorean Identity; 1+cot²x=1+2cotx+cot²x 4. Set equal to 0; 0=2cotx 5. Simplify 0=cotx 6.cot is x/y, x must be 0. x is 0 at π/2 and 3π/2 7. Check answers since you squared both sides; csc(π/2)-cot(π/2)=1 = 1-0= 1 ✓ csc(3π/2)-cot(3π/2)=1 = -1-0≠1 X x=(π/2)

9) solve for all values of x; 5cotx+4=4cotx+3

1. solve for cos; cosx=-1 2. cot is x/y. For x/y to be -1, x and y have to be the same numbers but opposite signs. They have to be (√2/2, √2/2) in quadrants 2 and 4 x=(3π/4)+2πn, (7π/4)+2πn

secθ

1/cosθ

tanθ

1/cotθ

sinθ

1/cscθ

cosθ

1/secθ

cscθ

1/sinθ

cotθ

1/tanθ

Double angle identity for cos2θ

cos2θ=cos²θ-sin²θ

cotθ quotient identity

cosθ/sinθ

Double angle Identity for sin2θ

sin2θ=2sinθcosθ

Pythagorean identity sin and cos

sin²θ+cos²θ=1

Pythagorean Identities

sin²θ+cos²θ=1 1+tan²θ=sec²θ 1+cot²θ=csc²θ

tanθ quotient identity

sinθ/cosθ

add 2πn

solve for all values of x

Quotient Identities

tanθ=(sinθ/cosθ) cotθ=(cosθ/sinθ)