To express the given function [tex]\( y = 9 \cos x + 5 \sin x \)[/tex] in the form [tex]\( y = R \cos(x - \theta) \)[/tex], we follow a detailed step-by-step approach.
1. Identify the coefficients from the given function:
[tex]\[
y = 9 \cos x + 5 \sin x
\][/tex]
Here, the coefficient of [tex]\(\cos x\)[/tex] is 9 and the coefficient of [tex]\(\sin x\)[/tex] is 5.
2. Calculate the amplitude [tex]\( R \)[/tex]:
The amplitude [tex]\( R \)[/tex] can be found using the formula:
[tex]\[
R = \sqrt{(a^2 + b^2)}
\][/tex]
where [tex]\( a \)[/tex] is 9 and [tex]\( b \)[/tex] is 5. Substituting the given values:
[tex]\[
R = \sqrt{(9^2 + 5^2)} = \sqrt{(81 + 25)} = \sqrt{106}
\][/tex]
3. Simplify the expression for [tex]\( R \)[/tex]:
[tex]\[
R = \sqrt{106} \approx 10.295630140987
\][/tex]
4. Calculate the phase shift [tex]\( \theta \)[/tex]:
The phase shift [tex]\( \theta \)[/tex] can be determined using the formula:
[tex]\[
\tan(\theta) = \frac{b}{a} = \frac{5}{9}
\][/tex]
Hence, [tex]\(\theta\)[/tex] is the arctangent of [tex]\( \frac{5}{9} \)[/tex]:
[tex]\[
\theta = \arctan(\frac{5}{9})
\][/tex]
5. Find the value of [tex]\( \theta \)[/tex] in radians:
[tex]\[
\theta = \arctan(\frac{5}{9}) \approx 0.507098504392337 \text{ radians}
\][/tex]
6. Combine both results to rewrite the function:
With [tex]\( R \approx 10.295630140987 \)[/tex] and [tex]\(\theta \approx 0.507098504392337\)[/tex], the function [tex]\( y \)[/tex] can be expressed as:
[tex]\[
y = R \cos(x - \theta) \approx 10.295630140987 \cos(x - 0.507098504392337)
\][/tex]
Therefore, the given function [tex]\( y = 9 \cos x + 5 \sin x \)[/tex] can be expressed in the simplified form as:
[tex]\[
y \approx 10.295630140987 \cos(x - 0.507098504392337)
\][/tex]
