To solve this problem, we need to find the value of $R$ in the expression:

$a \cos(\theta) + b \sin(\theta) = R \cos(\theta + \alpha)$

This equation can be transformed by applying the trigonometric identity for $R \cos(\theta + \alpha)$, which expands to:

$R \cos(\theta + \alpha) = R \cos(\alpha) \cos(\theta) - R \sin(\alpha) \sin(\theta)$

Matching terms, we get:

1. $R \cos(\alpha) = a$
2. $R \sin(\alpha) = b$

To find $R$, we use the Pythagorean identity:

$R = \sqrt{a^2 + b^2}$

Given:

• $a = 5.1$
• $b = 4.3$

Calculation

$R = \sqrt{(5.1)^2 + (4.3)^2}$ $R = \sqrt{26.01 + 18.49}$ $R = \sqrt{44.5}$ $R \approx 6.670$

Therefore, the value of $R$ to three decimal places is approximately $\mathbf{6.670}$.

Would you like further details, or do you have any questions?

Related Questions

1. How would you find the angle $\alpha$ in this problem?
2. What is the general approach to solve for $R$ and $\alpha$ in such trigonometric expressions?
3. How can we verify the identity for $R \cos(\theta + \alpha)$?
4. What is the significance of representing a trigonometric expression in terms of a single cosine term?
5. How does this approach relate to vector magnitude and direction?

Tip

When you encounter trigonometric expressions involving sums of sine and cosine terms, look for opportunities to simplify them using amplitude and phase shift techniques.