Answer :
To solve for \(\cos (\alpha + \beta)\), \(\sin (\alpha + \beta)\), and \(\tan (\alpha + \beta)\) given \(\sin (\alpha)=\frac{4}{5}\) where \(\alpha\) lies in quadrant I, and \(\sin (\beta)=\frac{24}{25}\) where \(\beta\) lies in quadrant II, let's proceed step-by-step.
### Step 1: Finding \(\cos(\alpha)\)
Since \(\alpha\) is in quadrant I, \(\cos(\alpha)\) will be positive.
We have:
[tex]\[ \sin(\alpha) = \frac{4}{5} \][/tex]
Using the Pythagorean identity \(\sin^2(\alpha) + \cos^2(\alpha) = 1\):
[tex]\[ \cos^2(\alpha) = 1 - \sin^2(\alpha) \][/tex]
[tex]\[ \cos^2(\alpha) = 1 - \left( \frac{4}{5} \right)^2 \][/tex]
[tex]\[ \cos^2(\alpha) = 1 - \frac{16}{25} \][/tex]
[tex]\[ \cos^2(\alpha) = \frac{9}{25} \][/tex]
[tex]\[ \cos(\alpha) = \sqrt{\frac{9}{25}} \][/tex]
[tex]\[ \cos(\alpha) = \frac{3}{5} \][/tex]
### Step 2: Finding \(\cos(\beta)\)
Since \(\beta\) is in quadrant II, \(\cos(\beta)\) will be negative.
We have:
[tex]\[ \sin(\beta) = \frac{24}{25} \][/tex]
Using the Pythagorean identity \(\sin^2(\beta) + \cos^2(\beta) = 1\):
[tex]\[ \cos^2(\beta) = 1 - \sin^2(\beta) \][/tex]
[tex]\[ \cos^2(\beta) = 1 - \left( \frac{24}{25} \right)^2 \][/tex]
[tex]\[ \cos^2(\beta) = 1 - \frac{576}{625} \][/tex]
[tex]\[ \cos^2(\beta) = \frac{49}{625} \][/tex]
[tex]\[ \cos(\beta) = -\sqrt{\frac{49}{625}} \][/tex]
[tex]\[ \cos(\beta) = -\frac{7}{25} \][/tex]
### Step 3: Using Angle Addition Formulas
#### Finding \(\cos (\alpha + \beta)\)
[tex]\[ \cos (\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \][/tex]
Substitute the values for \(\cos(\alpha)\), \(\cos(\beta)\), \(\sin(\alpha)\), and \(\sin(\beta)\):
[tex]\[ \cos (\alpha + \beta) = \left( \frac{3}{5} \right) \left( -\frac{7}{25} \right) - \left( \frac{4}{5} \right) \left( \frac{24}{25} \right) \][/tex]
[tex]\[ \cos (\alpha + \beta) = \left( \frac{3 \times -7}{125} \right) - \left( \frac{4 \times 24}{125} \right) \][/tex]
[tex]\[ \cos (\alpha + \beta) = \frac{-21}{125} - \frac{96}{125} \][/tex]
[tex]\[ \cos (\alpha + \beta) = \frac{-117}{125} \][/tex]
[tex]\[ \cos (\alpha + \beta) = -0.936 \][/tex]
So,
[tex]\[ \cos (\alpha + \beta) = - \boxed{0.936} \][/tex]
#### Finding \(\sin (\alpha + \beta)\)
[tex]\[ \sin (\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) \][/tex]
Substitute the values for \(\cos(\alpha)\), \(\cos(\beta)\), \(\sin(\alpha)\), and \(\sin(\beta)\):
[tex]\[ \sin (\alpha + \beta) = \left( \frac{4}{5} \right) \left( -\frac{7}{25} \right) + \left( \frac{3}{5} \right) \left( \frac{24}{25} \right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = \left( \frac{4 \times -7}{125} \right) + \left( \frac{3 \times 24}{125} \right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = \frac{-28}{125} + \frac{72}{125} \][/tex]
[tex]\[ \sin (\alpha + \beta) = \frac{44}{125} \][/tex]
[tex]\[ \sin (\alpha + \beta) = 0.352 \][/tex]
So,
[tex]\[ \sin (\alpha + \beta) = \boxed{0.352} \][/tex]
#### Finding \(\tan (\alpha + \beta)\)
[tex]\[ \tan (\alpha + \beta) = \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)} \][/tex]
Substitute the values for \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta)\):
[tex]\[ \tan (\alpha + \beta) = \frac{0.352}{-0.936} \][/tex]
[tex]\[ \tan (\alpha + \beta) = -0.376 \][/tex]
So,
[tex]\[ \tan (\alpha + \beta) = - \boxed{0.376} \][/tex]
These values match the results obtained from calculations:
[tex]\[ \cos (\alpha + \beta) = -0.936 \][/tex]
[tex]\[ \sin (\alpha + \beta) = 0.352 \][/tex]
[tex]\[ \tan (\alpha + \beta) = -0.376 \][/tex]
### Step 1: Finding \(\cos(\alpha)\)
Since \(\alpha\) is in quadrant I, \(\cos(\alpha)\) will be positive.
We have:
[tex]\[ \sin(\alpha) = \frac{4}{5} \][/tex]
Using the Pythagorean identity \(\sin^2(\alpha) + \cos^2(\alpha) = 1\):
[tex]\[ \cos^2(\alpha) = 1 - \sin^2(\alpha) \][/tex]
[tex]\[ \cos^2(\alpha) = 1 - \left( \frac{4}{5} \right)^2 \][/tex]
[tex]\[ \cos^2(\alpha) = 1 - \frac{16}{25} \][/tex]
[tex]\[ \cos^2(\alpha) = \frac{9}{25} \][/tex]
[tex]\[ \cos(\alpha) = \sqrt{\frac{9}{25}} \][/tex]
[tex]\[ \cos(\alpha) = \frac{3}{5} \][/tex]
### Step 2: Finding \(\cos(\beta)\)
Since \(\beta\) is in quadrant II, \(\cos(\beta)\) will be negative.
We have:
[tex]\[ \sin(\beta) = \frac{24}{25} \][/tex]
Using the Pythagorean identity \(\sin^2(\beta) + \cos^2(\beta) = 1\):
[tex]\[ \cos^2(\beta) = 1 - \sin^2(\beta) \][/tex]
[tex]\[ \cos^2(\beta) = 1 - \left( \frac{24}{25} \right)^2 \][/tex]
[tex]\[ \cos^2(\beta) = 1 - \frac{576}{625} \][/tex]
[tex]\[ \cos^2(\beta) = \frac{49}{625} \][/tex]
[tex]\[ \cos(\beta) = -\sqrt{\frac{49}{625}} \][/tex]
[tex]\[ \cos(\beta) = -\frac{7}{25} \][/tex]
### Step 3: Using Angle Addition Formulas
#### Finding \(\cos (\alpha + \beta)\)
[tex]\[ \cos (\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \][/tex]
Substitute the values for \(\cos(\alpha)\), \(\cos(\beta)\), \(\sin(\alpha)\), and \(\sin(\beta)\):
[tex]\[ \cos (\alpha + \beta) = \left( \frac{3}{5} \right) \left( -\frac{7}{25} \right) - \left( \frac{4}{5} \right) \left( \frac{24}{25} \right) \][/tex]
[tex]\[ \cos (\alpha + \beta) = \left( \frac{3 \times -7}{125} \right) - \left( \frac{4 \times 24}{125} \right) \][/tex]
[tex]\[ \cos (\alpha + \beta) = \frac{-21}{125} - \frac{96}{125} \][/tex]
[tex]\[ \cos (\alpha + \beta) = \frac{-117}{125} \][/tex]
[tex]\[ \cos (\alpha + \beta) = -0.936 \][/tex]
So,
[tex]\[ \cos (\alpha + \beta) = - \boxed{0.936} \][/tex]
#### Finding \(\sin (\alpha + \beta)\)
[tex]\[ \sin (\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) \][/tex]
Substitute the values for \(\cos(\alpha)\), \(\cos(\beta)\), \(\sin(\alpha)\), and \(\sin(\beta)\):
[tex]\[ \sin (\alpha + \beta) = \left( \frac{4}{5} \right) \left( -\frac{7}{25} \right) + \left( \frac{3}{5} \right) \left( \frac{24}{25} \right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = \left( \frac{4 \times -7}{125} \right) + \left( \frac{3 \times 24}{125} \right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = \frac{-28}{125} + \frac{72}{125} \][/tex]
[tex]\[ \sin (\alpha + \beta) = \frac{44}{125} \][/tex]
[tex]\[ \sin (\alpha + \beta) = 0.352 \][/tex]
So,
[tex]\[ \sin (\alpha + \beta) = \boxed{0.352} \][/tex]
#### Finding \(\tan (\alpha + \beta)\)
[tex]\[ \tan (\alpha + \beta) = \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)} \][/tex]
Substitute the values for \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta)\):
[tex]\[ \tan (\alpha + \beta) = \frac{0.352}{-0.936} \][/tex]
[tex]\[ \tan (\alpha + \beta) = -0.376 \][/tex]
So,
[tex]\[ \tan (\alpha + \beta) = - \boxed{0.376} \][/tex]
These values match the results obtained from calculations:
[tex]\[ \cos (\alpha + \beta) = -0.936 \][/tex]
[tex]\[ \sin (\alpha + \beta) = 0.352 \][/tex]
[tex]\[ \tan (\alpha + \beta) = -0.376 \][/tex]
