Answer :
Given that:
[tex]\[ \sin(\alpha) = \frac{5}{13}, \quad 0 < \alpha < \frac{\pi}{2} \][/tex]
[tex]\[ \cos(\beta) = \frac{4\sqrt{97}}{97}, \quad -\frac{\pi}{2} < \beta < 0 \][/tex]
We need to find the following values:
(a) [tex]\(\sin(\alpha + \beta)\)[/tex]
(b) [tex]\(\cos(\alpha + \beta)\)[/tex]
(c) [tex]\(\sin(\alpha - \beta)\)[/tex]
(d) [tex]\(\tan(\alpha - \beta)\)[/tex]
### (a) [tex]\(\sin(\alpha + \beta)\)[/tex]
First, we need to find [tex]\(\cos(\alpha)\)[/tex] and [tex]\(\sin(\beta)\)[/tex].
From [tex]\(\sin(\alpha)\)[/tex], we can find [tex]\(\cos(\alpha)\)[/tex] using the Pythagorean identity:
[tex]\[ \cos^2(\alpha) = 1 - \sin^2(\alpha) \][/tex]
[tex]\[ \cos(\alpha) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \][/tex]
Similarly, using the Pythagorean identity for [tex]\(\beta\)[/tex]:
[tex]\[ \sin^2(\beta) = 1 - \cos^2(\beta) \][/tex]
Since [tex]\(\beta\)[/tex] is in the fourth quadrant ([tex]\(-\frac{\pi}{2} < \beta < 0\)[/tex]), [tex]\(\sin(\beta)\)[/tex] should be negative:
[tex]\[ \sin(\beta) = -\sqrt{1 - \left(\frac{4\sqrt{97}}{97}\right)^2} = -\sqrt{1 - \frac{16 \cdot 97}{97^2}} = -\sqrt{1 - \frac{1552}{9409}} = -\sqrt{\frac{7857}{9409}} = -\frac{\sqrt{7857}}{97} \][/tex]
Now, we use the angle addition formula for sine:
[tex]\[ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \sin(\alpha + \beta) = \left(\frac{5}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) + \left(\frac{12}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \sin(\alpha + \beta) = \frac{20\sqrt{97}}{1261} - \frac{12\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \sin(\alpha + \beta) \approx -0.6873112502442961 \][/tex]
Thus:
[tex]\[ \sin(\alpha + \beta) = -0.6873112502442961 \][/tex]
### (b) [tex]\(\cos(\alpha + \beta)\)[/tex]
Using the angle addition formula for cosine:
[tex]\[ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \cos(\alpha + \beta) = \left(\frac{12}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{5}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \cos(\alpha + \beta) = \frac{48\sqrt{97}}{1261} + \frac{5\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \cos(\alpha + \beta) \approx 0.7263630258263583 \][/tex]
Thus:
[tex]\[ \cos(\alpha + \beta) = 0.7263630258263583 \][/tex]
### (c) [tex]\(\sin(\alpha - \beta)\)[/tex]
Using the angle subtraction formula for sine:
[tex]\[ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \sin(\alpha - \beta) = \left(\frac{5}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{12}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \sin(\alpha - \beta) = \frac{20\sqrt{97}}{1261} + \frac{12\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \sin(\alpha - \beta) \approx 0.9997254549007942 \][/tex]
Thus:
[tex]\[ \sin(\alpha - \beta) = 0.9997254549007942 \][/tex]
### (d) [tex]\(\tan(\alpha - \beta)\)[/tex]
Using the formula for tangent:
[tex]\[ \tan(\alpha - \beta) = \frac{\sin(\alpha - \beta)}{\cos(\alpha - \beta)} \][/tex]
Using the values from (c) and (b):
[tex]\[ \tan(\alpha - \beta) = \frac{0.9997254549007942}{0.7263630258263583} \][/tex]
Simplify this to get:
[tex]\[ \tan(\alpha - \beta) = 1.3763440860215055 \][/tex]
Thus:
[tex]\[ \tan(\alpha - \beta) = 1.3763440860215055 \][/tex]
[tex]\[ \sin(\alpha) = \frac{5}{13}, \quad 0 < \alpha < \frac{\pi}{2} \][/tex]
[tex]\[ \cos(\beta) = \frac{4\sqrt{97}}{97}, \quad -\frac{\pi}{2} < \beta < 0 \][/tex]
We need to find the following values:
(a) [tex]\(\sin(\alpha + \beta)\)[/tex]
(b) [tex]\(\cos(\alpha + \beta)\)[/tex]
(c) [tex]\(\sin(\alpha - \beta)\)[/tex]
(d) [tex]\(\tan(\alpha - \beta)\)[/tex]
### (a) [tex]\(\sin(\alpha + \beta)\)[/tex]
First, we need to find [tex]\(\cos(\alpha)\)[/tex] and [tex]\(\sin(\beta)\)[/tex].
From [tex]\(\sin(\alpha)\)[/tex], we can find [tex]\(\cos(\alpha)\)[/tex] using the Pythagorean identity:
[tex]\[ \cos^2(\alpha) = 1 - \sin^2(\alpha) \][/tex]
[tex]\[ \cos(\alpha) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \][/tex]
Similarly, using the Pythagorean identity for [tex]\(\beta\)[/tex]:
[tex]\[ \sin^2(\beta) = 1 - \cos^2(\beta) \][/tex]
Since [tex]\(\beta\)[/tex] is in the fourth quadrant ([tex]\(-\frac{\pi}{2} < \beta < 0\)[/tex]), [tex]\(\sin(\beta)\)[/tex] should be negative:
[tex]\[ \sin(\beta) = -\sqrt{1 - \left(\frac{4\sqrt{97}}{97}\right)^2} = -\sqrt{1 - \frac{16 \cdot 97}{97^2}} = -\sqrt{1 - \frac{1552}{9409}} = -\sqrt{\frac{7857}{9409}} = -\frac{\sqrt{7857}}{97} \][/tex]
Now, we use the angle addition formula for sine:
[tex]\[ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \sin(\alpha + \beta) = \left(\frac{5}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) + \left(\frac{12}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \sin(\alpha + \beta) = \frac{20\sqrt{97}}{1261} - \frac{12\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \sin(\alpha + \beta) \approx -0.6873112502442961 \][/tex]
Thus:
[tex]\[ \sin(\alpha + \beta) = -0.6873112502442961 \][/tex]
### (b) [tex]\(\cos(\alpha + \beta)\)[/tex]
Using the angle addition formula for cosine:
[tex]\[ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \cos(\alpha + \beta) = \left(\frac{12}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{5}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \cos(\alpha + \beta) = \frac{48\sqrt{97}}{1261} + \frac{5\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \cos(\alpha + \beta) \approx 0.7263630258263583 \][/tex]
Thus:
[tex]\[ \cos(\alpha + \beta) = 0.7263630258263583 \][/tex]
### (c) [tex]\(\sin(\alpha - \beta)\)[/tex]
Using the angle subtraction formula for sine:
[tex]\[ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \][/tex]
Substituting the known values:
[tex]\[ \sin(\alpha - \beta) = \left(\frac{5}{13}\right)\left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{12}{13}\right)\left(-\frac{\sqrt{7857}}{97}\right) \][/tex]
Simplify this to get:
[tex]\[ \sin(\alpha - \beta) = \frac{20\sqrt{97}}{1261} + \frac{12\sqrt{7857}}{1261} \][/tex]
Using the numerical result:
[tex]\[ \sin(\alpha - \beta) \approx 0.9997254549007942 \][/tex]
Thus:
[tex]\[ \sin(\alpha - \beta) = 0.9997254549007942 \][/tex]
### (d) [tex]\(\tan(\alpha - \beta)\)[/tex]
Using the formula for tangent:
[tex]\[ \tan(\alpha - \beta) = \frac{\sin(\alpha - \beta)}{\cos(\alpha - \beta)} \][/tex]
Using the values from (c) and (b):
[tex]\[ \tan(\alpha - \beta) = \frac{0.9997254549007942}{0.7263630258263583} \][/tex]
Simplify this to get:
[tex]\[ \tan(\alpha - \beta) = 1.3763440860215055 \][/tex]
Thus:
[tex]\[ \tan(\alpha - \beta) = 1.3763440860215055 \][/tex]
