$$ \frac { \sin 20 \alpha + \sin 44 \alpha + \sin 8 \alpha + \sin 2 \alpha } { \cos 2 \alpha + \cos 14 \alpha + \cos 8 \alpha + \cos 2 \alpha } = an 44 $$

Question

$$ \frac { \sin 20 \alpha + \sin 44 \alpha + \sin 8 \alpha + \sin 2 \alpha } { \cos 2 \alpha + \cos 14 \alpha + \cos 8 \alpha + \cos 2 \alpha } = 	an 44 $$

UpStudy AI · Accepted Answer

To solve the equation

$$
\frac { \sin 20 \alpha + \sin 44 \alpha + \sin 8 \alpha + \sin 2 \alpha } { \cos 2 \alpha + \cos 14 \alpha + \cos 8 \alpha + \cos 2 \alpha } = \tan 44,
$$

we will first simplify the numerator and the denominator separately.

### Step 1: Simplifying the Numerator

The numerator is

$$
\sin 20 \alpha + \sin 44 \alpha + \sin 8 \alpha + \sin 2 \alpha.
$$

### Step 2: Simplifying the Denominator

The denominator is

$$
\cos 2 \alpha + \cos 14 \alpha + \cos 8 \alpha + \cos 2 \alpha.
$$

### Step 3: Setting Up the Equation

We will set the simplified numerator and denominator equal to $$ \tan 44 $$.

Let's calculate the numerator and denominator using trigonometric identities.

1. **Numerator**: We can use the sum-to-product identities to combine terms.
2. **Denominator**: Similarly, we can apply the sum-to-product identities.

Let's proceed with these calculations.

I'll first calculate the numerator and denominator separately.
Simplify the expression by following steps:
- step0: Solution:
  $$\cos\left(2alpha\right)+\cos\left(14alpha\right)+\cos\left(8alpha\right)+\cos\left(2alpha\right)$$
- step1: Multiply the terms:
  $$\cos\left(2a^{2}lph\right)+\cos\left(14alpha\right)+\cos\left(8alpha\right)+\cos\left(2alpha\right)$$
- step2: Multiply the terms:
  $$\cos\left(2a^{2}lph\right)+\cos\left(14a^{2}lph\right)+\cos\left(8alpha\right)+\cos\left(2alpha\right)$$
- step3: Multiply the terms:
  $$\cos\left(2a^{2}lph\right)+\cos\left(14a^{2}lph\right)+\cos\left(8a^{2}lph\right)+\cos\left(2alpha\right)$$
- step4: Multiply the terms:
  $$\cos\left(2a^{2}lph\right)+\cos\left(14a^{2}lph\right)+\cos\left(8a^{2}lph\right)+\cos\left(2a^{2}lph\right)$$
- step5: Transform the expression:
  $$\cos\left(2a^{2}lph\right)+2\cos\left(11a^{2}lph\right)\cos\left(3a^{2}lph\right)+\cos\left(2a^{2}lph\right)$$
- step6: Add the terms:
  $$2\cos\left(2a^{2}lph\right)+2\cos\left(11a^{2}lph\right)\cos\left(3a^{2}lph\right)$$
- step7: Transform the expression:
  $$2\cos\left(2a^{2}lph\right)+2\cos\left(3a^{2}lph\right)\cos\left(11a^{2}lph\right)$$
- step8: Rewrite the expression:
  $$2\cos\left(2a^{2}lph\right)+\cos\left(14a^{2}lph\right)+\cos\left(8a^{2}lph\right)$$
Calculate or simplify the expression $$ \sin(20*alpha) + \sin(44*alpha) + \sin(8*alpha) + \sin(2*alpha) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\sin\left(20alpha\right)+\sin\left(44alpha\right)+\sin\left(8alpha\right)+\sin\left(2alpha\right)$$
- step1: Multiply the terms:
  $$\sin\left(20a^{2}lph\right)+\sin\left(44alpha\right)+\sin\left(8alpha\right)+\sin\left(2alpha\right)$$
- step2: Multiply the terms:
  $$\sin\left(20a^{2}lph\right)+\sin\left(44a^{2}lph\right)+\sin\left(8alpha\right)+\sin\left(2alpha\right)$$
- step3: Multiply the terms:
  $$\sin\left(20a^{2}lph\right)+\sin\left(44a^{2}lph\right)+\sin\left(8a^{2}lph\right)+\sin\left(2alpha\right)$$
- step4: Multiply the terms:
  $$\sin\left(20a^{2}lph\right)+\sin\left(44a^{2}lph\right)+\sin\left(8a^{2}lph\right)+\sin\left(2a^{2}lph\right)$$
- step5: Transform the expression:
  $$2\sin\left(14a^{2}lph\right)\cos\left(6a^{2}lph\right)+\sin\left(44a^{2}lph\right)+\sin\left(2a^{2}lph\right)$$
- step6: Transform the expression:
  $$2\sin\left(14a^{2}lph\right)\cos\left(6a^{2}lph\right)+2\sin\left(23a^{2}lph\right)\cos\left(21a^{2}lph\right)$$
- step7: Transform the expression:
  $$\sin\left(20a^{2}lph\right)+\sin\left(8a^{2}lph\right)+2\sin\left(23a^{2}lph\right)\cos\left(21a^{2}lph\right)$$
- step8: Transform the expression:
  $$\sin\left(20a^{2}lph\right)+\sin\left(8a^{2}lph\right)+\sin\left(44a^{2}lph\right)+\sin\left(2a^{2}lph\right)$$
The calculations for the numerator and denominator yield:

### Numerator:
$$
\sin(20\alpha) + \sin(44\alpha) + \sin(8\alpha) + \sin(2\alpha)
$$

### Denominator:
$$
2\cos(2\alpha) + \cos(14\alpha) + \cos(8\alpha)
$$

Now, we can set up the equation:

$$
\frac{\sin(20\alpha) + \sin(44\alpha) + \sin(8\alpha) + \sin(2\alpha)}{2\cos(2\alpha) + \cos(14\alpha) + \cos(8\alpha)} = \tan(44)
$$

Next, we can analyze this equation further or substitute specific values for $$\alpha$$ to find solutions.

Would you like to proceed with a specific value for $$\alpha$$ or continue simplifying the equation?
To solve the equation $$ \frac{\sin 20\alpha + \sin 44\alpha + \sin 8\alpha + \sin 2\alpha}{\cos 2\alpha + \cos 14\alpha + \cos 8\alpha + \cos 2\alpha} = \tan 44, $$ we can simplify the numerator and denominator using trigonometric identities. After simplification, the equation becomes: $$ \frac{\sin 20\alpha + \sin 44\alpha + \sin 8\alpha + \sin 2\alpha}{2\cos 2\alpha + \cos 14\alpha + \cos 8\alpha} = \tan 44. $$ This equation can be further analyzed or solved by substituting specific values for $$\alpha$$ to find solutions.

Upstudy AI Solution

Answer

Solution

Step 1: Simplifying the Numerator

Step 2: Simplifying the Denominator

Step 3: Setting Up the Equation

Numerator:

Denominator:

Beyond the Answer

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