5.3. Calculate $$ 2 \sin heta+\cos heta $$ without the use of a calculator.

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5.3. Calculate $$ 2 \sin 	heta+\cos 	heta $$ without the use of a calculator.

UpStudy AI · Accepted Answer

To calculate $$ 2 \sin \theta + \cos \theta $$ without using a calculator, we can express the expression in the form of a single sine function using the **amplitude-phase** method. This approach simplifies the combination of sine and cosine terms into a single sine term with a phase shift.

### Step 1: Express as a Single Sine Function

We aim to rewrite $$ 2 \sin \theta + \cos \theta $$ in the form:
$$
R \sin(\theta + \alpha)
$$
where:
- $$ R $$ is the amplitude.
- $$ \alpha $$ is the phase shift.

### Step 2: Determine Amplitude $$ R $$ and Phase Shift $$ \alpha $$

Using the **angle addition formula** for sine:
$$
R \sin(\theta + \alpha) = R \sin \theta \cos \alpha + R \cos \theta \sin \alpha
$$
Compare this with the original expression:
$$
2 \sin \theta + \cos \theta
$$
By equating the coefficients, we get:
$$
R \cos \alpha = 2 \quad \text{(1)}
$$
$$
R \sin \alpha = 1 \quad \text{(2)}
$$

To find $$ R $$, square and add equations (1) and (2):
$$
(R \cos \alpha)^2 + (R \sin \alpha)^2 = 2^2 + 1^2
$$
$$
R^2 (\cos^2 \alpha + \sin^2 \alpha) = 5
$$
$$
R^2 = 5 \quad \Rightarrow \quad R = \sqrt{5}
$$

Next, find $$ \alpha $$ using equation (2):
$$
\sin \alpha = \frac{1}{\sqrt{5}} \quad \Rightarrow \quad \alpha = \arcsin\left(\frac{1}{\sqrt{5}}\right)
$$
Alternatively, since $$ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{1}{2} $$, we can write:
$$
\alpha = \arctan\left(\frac{1}{2}\right)
$$

### Step 3: Final Expression

Substituting $$ R $$ and $$ \alpha $$ back into the single sine function form:
$$
2 \sin \theta + \cos \theta = \sqrt{5} \sin\left(\theta + \arctan\left(\frac{1}{2}\right)\right)
$$

This expression represents $$ 2 \sin \theta + \cos \theta $$ as a single sine function with amplitude $$ \sqrt{5} $$ and phase shift $$ \arctan\left(\frac{1}{2}\right) $$.

### Conclusion

Therefore, the expression $$ 2 \sin \theta + \cos \theta $$ can be rewritten without a calculator as:
$$
2 \sin \theta + \cos \theta = \sqrt{5} \sin\left(\theta + \arctan\left(\frac{1}{2}\right)\right)
$$

**Answer:**

After simplifying, the expression is equal to √5 times sin (θ plus arctan ½). Thus,

2 sinθ + cosθ = √5 sin( θ + arctan(1/2) )
$$ 2 \sin \theta + \cos \theta = \sqrt{5} \sin\left(\theta + \arctan\left(\frac{1}{2}\right)\right) $$

5.3. Calculate \( 2 \sin \theta+\cos \theta \) without the use of a calculator.

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