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To find the exact values of the trigonometric functions given the conditions [tex]\( \sin \alpha = \frac{4}{5} \)[/tex] where [tex]\( 0 < \alpha < \frac{\pi}{2} \)[/tex] (meaning [tex]\(\alpha\)[/tex] is in the first quadrant) and [tex]\( \cos \beta = \frac{4\sqrt{97}}{97} \)[/tex] where [tex]\(-\frac{\pi}{2} < \beta < 0\)[/tex] (meaning [tex]\(\beta\)[/tex] is in the fourth quadrant), we will use the angle addition and subtraction formulas.
Step-by-Step Solution:
Firstly, let's determine [tex]\(\cos \alpha\)[/tex] and [tex]\(\sin \beta\)[/tex]:
1. Calculate [tex]\(\cos \alpha\)[/tex]:
We know [tex]\( \sin^2 \alpha + \cos^2 \alpha = 1 \)[/tex].
Given [tex]\( \sin \alpha = \frac{4}{5} \)[/tex],
[tex]\[ \sin^2 \alpha = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \][/tex]
Thus,
[tex]\[ \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \frac{16}{25} = \frac{9}{25} \][/tex]
Since [tex]\( 0 < \alpha < \frac{\pi}{2} \)[/tex],
[tex]\[ \cos \alpha = \sqrt{\frac{9}{25}} = \frac{3}{5} \][/tex]
2. Calculate [tex]\(\sin \beta\)[/tex]:
We know [tex]\( \sin^2 \beta + \cos^2 \beta = 1 \)[/tex].
Given [tex]\( \cos \beta = \frac{4\sqrt{97}}{97} \)[/tex],
[tex]\[ \cos^2 \beta = \left(\frac{4\sqrt{97}}{97}\right)^2 = \frac{16 \cdot 97}{97^2} = \frac{16}{97} \][/tex]
Thus,
[tex]\[ \sin^2 \beta = 1 - \cos^2 \beta = 1 - \frac{16}{97} = \frac{81}{97} \][/tex]
Since [tex]\( -\frac{\pi}{2} < \beta < 0 \)[/tex], [tex]\(\beta\)[/tex] is in the fourth quadrant, where sine is negative,
[tex]\[ \sin \beta = -\sqrt{\frac{81}{97}} = -\frac{9}{\sqrt{97}} \][/tex]
Now we proceed to find the required values:
(a) [tex]\(\sin (\alpha + \beta)\)[/tex]:
[tex]\[ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \sin (\alpha + \beta) = \left(\frac{4}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) + \left(\frac{3}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \sin (\alpha + \beta) \approx -0.22337615632939606 \][/tex]
(b) [tex]\(\cos (\alpha + \beta)\)[/tex]:
[tex]\[ \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \cos (\alpha + \beta) = \left(\frac{3}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{4}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \cos (\alpha + \beta) \approx 0.9747323185282742 \][/tex]
(c) [tex]\(\sin (\alpha - \beta)\)[/tex]:
[tex]\[ \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \sin (\alpha - \beta) = \left(\frac{4}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{3}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \sin (\alpha - \beta) \approx 0.8731977020149122 \][/tex]
(d) [tex]\(\tan (\alpha - \beta)\)[/tex]:
We know,
[tex]\[ \tan (\alpha - \beta) = \frac{\sin (\alpha - \beta)}{\cos (\alpha - \beta)} \][/tex]
Using the values from parts (b) and (c),
[tex]\[ \cos (\alpha - \beta) = \cos (\alpha + \beta) \][/tex]
[tex]\[ \tan (\alpha - \beta) = \frac{\sin (\alpha - \beta)}{\cos (\alpha + \beta)} = \frac{0.8731977020149122}{0.9747323185282742} \][/tex]
After simplification, we get
[tex]\[ \tan (\alpha - \beta) \approx 0.8958333333333331 \][/tex]
Thus, the exact values are:
(a) [tex]\(\sin (\alpha + \beta) \approx -0.22337615632939606\)[/tex]
(b) [tex]\(\cos (\alpha + \beta) \approx 0.9747323185282742\)[/tex]
(c) [tex]\(\sin (\alpha - \beta) \approx 0.8731977020149122\)[/tex]
(d) [tex]\(\tan (\alpha - \beta) \approx 0.8958333333333331\)[/tex]
Step-by-Step Solution:
Firstly, let's determine [tex]\(\cos \alpha\)[/tex] and [tex]\(\sin \beta\)[/tex]:
1. Calculate [tex]\(\cos \alpha\)[/tex]:
We know [tex]\( \sin^2 \alpha + \cos^2 \alpha = 1 \)[/tex].
Given [tex]\( \sin \alpha = \frac{4}{5} \)[/tex],
[tex]\[ \sin^2 \alpha = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \][/tex]
Thus,
[tex]\[ \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \frac{16}{25} = \frac{9}{25} \][/tex]
Since [tex]\( 0 < \alpha < \frac{\pi}{2} \)[/tex],
[tex]\[ \cos \alpha = \sqrt{\frac{9}{25}} = \frac{3}{5} \][/tex]
2. Calculate [tex]\(\sin \beta\)[/tex]:
We know [tex]\( \sin^2 \beta + \cos^2 \beta = 1 \)[/tex].
Given [tex]\( \cos \beta = \frac{4\sqrt{97}}{97} \)[/tex],
[tex]\[ \cos^2 \beta = \left(\frac{4\sqrt{97}}{97}\right)^2 = \frac{16 \cdot 97}{97^2} = \frac{16}{97} \][/tex]
Thus,
[tex]\[ \sin^2 \beta = 1 - \cos^2 \beta = 1 - \frac{16}{97} = \frac{81}{97} \][/tex]
Since [tex]\( -\frac{\pi}{2} < \beta < 0 \)[/tex], [tex]\(\beta\)[/tex] is in the fourth quadrant, where sine is negative,
[tex]\[ \sin \beta = -\sqrt{\frac{81}{97}} = -\frac{9}{\sqrt{97}} \][/tex]
Now we proceed to find the required values:
(a) [tex]\(\sin (\alpha + \beta)\)[/tex]:
[tex]\[ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \sin (\alpha + \beta) = \left(\frac{4}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) + \left(\frac{3}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \sin (\alpha + \beta) \approx -0.22337615632939606 \][/tex]
(b) [tex]\(\cos (\alpha + \beta)\)[/tex]:
[tex]\[ \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \cos (\alpha + \beta) = \left(\frac{3}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{4}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \cos (\alpha + \beta) \approx 0.9747323185282742 \][/tex]
(c) [tex]\(\sin (\alpha - \beta)\)[/tex]:
[tex]\[ \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \][/tex]
Substituting the known values,
[tex]\[ \sin (\alpha - \beta) = \left(\frac{4}{5}\right) \left(\frac{4\sqrt{97}}{97}\right) - \left(\frac{3}{5}\right) \left(-\frac{9}{\sqrt{97}}\right) \][/tex]
After simplification, we get
[tex]\[ \sin (\alpha - \beta) \approx 0.8731977020149122 \][/tex]
(d) [tex]\(\tan (\alpha - \beta)\)[/tex]:
We know,
[tex]\[ \tan (\alpha - \beta) = \frac{\sin (\alpha - \beta)}{\cos (\alpha - \beta)} \][/tex]
Using the values from parts (b) and (c),
[tex]\[ \cos (\alpha - \beta) = \cos (\alpha + \beta) \][/tex]
[tex]\[ \tan (\alpha - \beta) = \frac{\sin (\alpha - \beta)}{\cos (\alpha + \beta)} = \frac{0.8731977020149122}{0.9747323185282742} \][/tex]
After simplification, we get
[tex]\[ \tan (\alpha - \beta) \approx 0.8958333333333331 \][/tex]
Thus, the exact values are:
(a) [tex]\(\sin (\alpha + \beta) \approx -0.22337615632939606\)[/tex]
(b) [tex]\(\cos (\alpha + \beta) \approx 0.9747323185282742\)[/tex]
(c) [tex]\(\sin (\alpha - \beta) \approx 0.8731977020149122\)[/tex]
(d) [tex]\(\tan (\alpha - \beta) \approx 0.8958333333333331\)[/tex]
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