Answer :
To find the exact value of [tex]\(\cos(\alpha + \beta)\)[/tex] where [tex]\(\cos \alpha = \frac{11}{61}\)[/tex] and [tex]\(\cos \beta = \frac{12}{37}\)[/tex], and given that the terminal side of [tex]\(\alpha\)[/tex] lies in quadrant IV and the terminal side of [tex]\(\beta\)[/tex] lies in quadrant I, we follow these steps:
1. Determine [tex]\(\sin \alpha\)[/tex] and [tex]\(\sin \beta\)[/tex]:
- Since [tex]\(\alpha\)[/tex] is in quadrant IV, [tex]\(\sin \alpha < 0\)[/tex].
- Since [tex]\(\beta\)[/tex] is in quadrant I, [tex]\(\sin \beta > 0\)[/tex].
2. Calculate [tex]\(\sin \alpha\)[/tex]:
[tex]\[ \cos^2 \alpha + \sin^2 \alpha = 1 \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \cos^2 \alpha \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \left(\frac{11}{61}\right)^2 \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \frac{121}{3721} \][/tex]
[tex]\[ \sin^2 \alpha = \frac{3721 - 121}{3721} \][/tex]
[tex]\[ \sin^2 \alpha = \frac{3600}{3721} \][/tex]
[tex]\[ \sin \alpha = -\sqrt{\frac{3600}{3721}} = -\frac{60}{61} \][/tex]
3. Calculate [tex]\(\sin \beta\)[/tex]:
[tex]\[ \cos^2 \beta + \sin^2 \beta = 1 \][/tex]
[tex]\[ \sin^2 \beta = 1 - \cos^2 \beta \][/tex]
[tex]\[ \sin^2 \beta = 1 - \left(\frac{12}{37}\right)^2 \][/tex]
[tex]\[ \sin^2 \beta = 1 - \frac{144}{1369} \][/tex]
[tex]\[ \sin^2 \beta = \frac{1369 - 144}{1369} \][/tex]
[tex]\[ \sin^2 \beta = \frac{1225}{1369} \][/tex]
[tex]\[ \sin \beta = \sqrt{\frac{1225}{1369}} = \frac{35}{37} \][/tex]
4. Use the angle addition formula for cosine:
[tex]\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \][/tex]
Substitute the known values:
[tex]\[ \cos(\alpha + \beta) = \left(\frac{11}{61}\right) \left(\frac{12}{37}\right) - \left(-\frac{60}{61}\right) \left(\frac{35}{37}\right) \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{11 \cdot 12}{61 \cdot 37} + \frac{60 \cdot 35}{61 \cdot 37} \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{132}{2257} + \frac{2100}{2257} \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{2232}{2257} \][/tex]
So, the exact value of [tex]\(\cos(\alpha + \beta)\)[/tex] is:
[tex]\[ \cos(\alpha + \beta) = \frac{2232}{2257} \][/tex]
1. Determine [tex]\(\sin \alpha\)[/tex] and [tex]\(\sin \beta\)[/tex]:
- Since [tex]\(\alpha\)[/tex] is in quadrant IV, [tex]\(\sin \alpha < 0\)[/tex].
- Since [tex]\(\beta\)[/tex] is in quadrant I, [tex]\(\sin \beta > 0\)[/tex].
2. Calculate [tex]\(\sin \alpha\)[/tex]:
[tex]\[ \cos^2 \alpha + \sin^2 \alpha = 1 \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \cos^2 \alpha \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \left(\frac{11}{61}\right)^2 \][/tex]
[tex]\[ \sin^2 \alpha = 1 - \frac{121}{3721} \][/tex]
[tex]\[ \sin^2 \alpha = \frac{3721 - 121}{3721} \][/tex]
[tex]\[ \sin^2 \alpha = \frac{3600}{3721} \][/tex]
[tex]\[ \sin \alpha = -\sqrt{\frac{3600}{3721}} = -\frac{60}{61} \][/tex]
3. Calculate [tex]\(\sin \beta\)[/tex]:
[tex]\[ \cos^2 \beta + \sin^2 \beta = 1 \][/tex]
[tex]\[ \sin^2 \beta = 1 - \cos^2 \beta \][/tex]
[tex]\[ \sin^2 \beta = 1 - \left(\frac{12}{37}\right)^2 \][/tex]
[tex]\[ \sin^2 \beta = 1 - \frac{144}{1369} \][/tex]
[tex]\[ \sin^2 \beta = \frac{1369 - 144}{1369} \][/tex]
[tex]\[ \sin^2 \beta = \frac{1225}{1369} \][/tex]
[tex]\[ \sin \beta = \sqrt{\frac{1225}{1369}} = \frac{35}{37} \][/tex]
4. Use the angle addition formula for cosine:
[tex]\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \][/tex]
Substitute the known values:
[tex]\[ \cos(\alpha + \beta) = \left(\frac{11}{61}\right) \left(\frac{12}{37}\right) - \left(-\frac{60}{61}\right) \left(\frac{35}{37}\right) \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{11 \cdot 12}{61 \cdot 37} + \frac{60 \cdot 35}{61 \cdot 37} \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{132}{2257} + \frac{2100}{2257} \][/tex]
[tex]\[ \cos(\alpha + \beta) = \frac{2232}{2257} \][/tex]
So, the exact value of [tex]\(\cos(\alpha + \beta)\)[/tex] is:
[tex]\[ \cos(\alpha + \beta) = \frac{2232}{2257} \][/tex]
