If `theta` and `phi` are acute angle. If `costheta=(3)/(5)` and `cosphi=(4)/(5)` then `cos(2theta+3phi)=?`

Options

A. `(4)/(5)`
B. `-(4)/(5)`
C. `(3)/(5)`
D. `-(3)/(5)`

Answer

To solve for \( \cos(2\theta + 3\phi) \) given that \( \cos \theta = \frac{3}{5} \) and \( \cos \phi = \frac{4}{5} \), we can follow these steps:

### Step 1: Find \( \sin \theta \) and \( \sin \phi \)

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

1. For \( \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

2. For \( \phi \):
\[ \sin^2 \phi = 1 - \cos^2 \phi = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ \sin \phi = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 2: Use the angle addition formula for cosine

We can express \( \cos(2\theta + 3\phi) \) using the cosine addition formula:
\[ \cos(2\theta + 3\phi) = \cos(2\theta + 2\phi + \phi) = \cos(2\theta + 2\phi)\cos(\phi) - \sin(2\theta + 2\phi)\sin(\phi) \]

### Step 3: Calculate \( \cos(2\theta) \) and \( \sin(2\theta) \)

Using the double angle formulas:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 = 2\left(\frac{3}{5}\right)^2 - 1 = 2\left(\frac{9}{25}\right) - 1 = \frac{18}{25} - 1 = \frac{-7}{25} \]
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2\left(\frac{4}{5}\right)\left(\frac{3}{5}\right) = \frac{24}{25} \]

### Step 4: Calculate \( \cos(2\phi) \) and \( \sin(2\phi) \)

Using the double angle formulas:
\[ \cos(2\phi) = 2\cos^2(\phi) - 1 = 2\left(\frac{4}{5}\right)^2 - 1 = 2\left(\frac{16}{25}\right) - 1 = \frac{32}{25} - 1 = \frac{7}{25} \]
\[ \sin(2\phi) = 2\sin(\phi)\cos(\phi) = 2\left(\frac{3}{5}\right)\left(\frac{4}{5}\right) = \frac{24}{25} \]

### Step 5: Substitute into the cosine addition formula

Now substituting back into the cosine addition formula:
\[ \cos(2\theta + 3\phi) = \cos(2\theta + 2\phi)\cos(\phi) - \sin(2\theta + 2\phi)\sin(\phi) \]

Calculating \( \cos(2\theta + 2\phi) \):
\[ \cos(2\theta + 2\phi) = \cos(2\theta)\cos(2\phi) - \sin(2\theta)\sin(2\phi) \]
\[ = \left(\frac{-7}{25}\right)\left(\frac{7}{25}\right) - \left(\frac{24}{25}\right)\left(\frac{24}{25}\right) \]
\[ = \frac{-49}{625} - \frac{576}{625} = \frac{-625}{625} = -1 \]

Now substituting back:
\[ \cos(2\theta + 3\phi) = (-1) \cdot \frac{4}{5} - \frac{24}{25} \cdot \frac{3}{5} \]
\[ = -\frac{4}{5} - \frac{72}{125} \]
Finding a common denominator (125):
\[ = -\frac{100}{125} - \frac{72}{125} = -\frac{172}{125} \]

### Final Answer
Thus, the final answer is:
\[ \cos(2\theta + 3\phi) = -\frac{172}{125} \]