Question

(a) Express \( 5 \sin x-3 \cos x \) in the form \( R \sin (x-\alpha) \), where \( R>0 \) and \( 0<\alpha<\frac{1}{2} \pi \). Give the exact value of \( R \) and give \( \alpha \) correct to 2 decimal places. [3] (b) Hence state the greatest and least possible values of \( (5 \sin x-3 \cos x)^{2} \). [2]

          (a) Express \( 5 \sin x-3 \cos x \) in the form \( R \sin (x-\alpha) \), where \( R>0 \) and \( 0<\alpha<\frac{1}{2} \pi \). Give the exact value of \( R \) and give \( \alpha \) correct to 2 decimal places.
[3]
(b) Hence state the greatest and least possible values of \( (5 \sin x-3 \cos x)^{2} \).
[2]

(a) Express 5 sin x-3 cos x in the form R sin (x-α), where R>0 and 0<α<(1)/(2)π. Give the exact value of R and give α correct to 2 decimal places.
[3]
(b) Hence state the greatest and least possible values of (5 sin x-3 cos x)^2.
[2]

Added by William H.

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan, Michael Sullivan, lll 3rd Edition

Chapter 10

Instant Answer

Solved by Expert Darshan Maheshwari

12/26/2021

Step 1

Step 1: To express \( 5 \sin x - 3 \cos x \) in the form \( R \sin (x - \alpha) \), we start by using the angle addition formula for sine: \[ R \sin (x - \alpha) = R (\sin x \cos \alpha - \cos x \sin \alpha) \] Show more…

Show all steps

Thanks for your feedback!

(a) Express \( 5 \sin x-3 \cos x \) in the form \( R \sin (x-\alpha) \), where \( R>0 \) and \( 0<\alpha<\frac{1}{2} \pi \). Give the exact value of \( R \) and give \( \alpha \) correct to 2 decimal places. [3] (b) Hence state the greatest and least possible values of \( (5 \sin x-3 \cos x)^{2} \). [2]