Question #252710

Express 2cosθ\theta +sinθ\theta in the form of Rcos(θ\thetaα\alpha) where r > 0 and 0o< α\alpha <90o.

Hence,

(a) find its maximum and minimum values.

(b) solve 2cosθ\theta +sinθ\theta = 2 for 0o less than or equal to θ\theta, θ\theta is less than or equal 360o. Give your answer to the nearest 0.1o.


1
Expert's answer
2021-10-19T17:09:02-0400
Rcos(θα)=Rcosθcosα+Rsinθsinα.Rcos(\theta-\alpha) =Rcos\theta cos\alpha + Rsin\theta sin\alpha.

2cosθ+sinθ=Rcosθcosα+Rsinθsinα.2cos\theta + sin\theta=Rcosθcosα+Rsinθsinα.

Rcosθcosα=2cosθ

Rsinθsinα=sinθ

Rcosα=2... eqtn 1

Rsinα=1...eqtn 2

Divide 2 by 1

Sinα/Cosα=1/2

Tanα=1/2

α=26.3o

Using SOH,CAH, TOA

Sinα=1/R

Cosα=2/R

Therefore R can be the hypotenuse

Hence, R=√ (12+22)

R=√5

2cosθ+sinθ=(√5)Cos(θ-26.3).

a)Using Rattafication formula for 2cosθ+sinθ

±√ (a2+ b2), where a =1, b =2, max is +and minimum is -.

Max is =+√(12+22)=√(-5).

Min is= - √(12+22)= -√(-5).


b) 2cosθ+sinθ=2

2cosθ-2=-sinθ

Square both sides

(2cosθ-2)2 =(-sinθ)2

4cos2θ -8cosθ+4=sin2θ

Recall that sin2θ=1-cos2θ

4cos2θ-8cosθ+4=1+cos2θ

4cos2θ-8cosθ+4-1-cos2θ=0

3cos2θ-8cosθ+3=0.

Using quadratic formula,

Cosθ=(-b±√(b2−4ac)) /2a

Where a=3,b=-8 and c=3.

Cosθ=(-(-8)±√((-8)2-(4x3x3))/(2x3)

Cosθ=(8±√(64-36))/6

Cosθ=(8±√(28)) /6

Cosθ=(8±5.29)/6

Cosθ=(8+5.29)/6 and (8-5.29)/6

Cosθ=(13.29/6) and (2.71/6)

Cosθ=2.22 and 0.45

Cosθ=2.22,Cosθ=0.45

Cosθ cannot be equal to 2.22.

Therefore Cosθ =0.45,θ=63.3o

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS