Let's find the angles [tex]$\theta$[/tex] that have the same sine and cosine values as [tex]$98^{\circ}$[/tex] within the range [tex]$0^{\circ} < \theta < 360^{\circ}$[/tex].
### Finding the angle with the same sine value as [tex]$98^{\circ}$[/tex]
The sine function is positive in both the first and second quadrants. For an angle [tex]\( \alpha \)[/tex] in the first quadrant, the corresponding angle in the second quadrant that has the same sine value is found by subtracting the given angle from [tex]$180^{\circ}$[/tex]:
[tex]\[ \sin(\theta) = \sin(98^{\circ}) \][/tex]
To find the missing angle [tex]$\theta$[/tex]:
[tex]\[ \theta = 180^{\circ} - 98^{\circ} \][/tex]
[tex]\[ \theta = 82^{\circ} \][/tex]
So, the angle with the same sine value as [tex]$98^{\circ}$[/tex] is [tex]\( \boxed{82^{\circ}} \)[/tex].
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### Finding the angle with the same cosine value as [tex]$98^{\circ}$[/tex]
The cosine function is negative in the second and third quadrants. For an angle [tex]\( \alpha \)[/tex] in the second quadrant, the corresponding angle in the fourth quadrant that has the same cosine value is found by subtracting the given angle from [tex]$360^{\circ}$[/tex]:
[tex]\[ \cos(\theta) = \cos(98^{\circ}) \][/tex]
To find the missing angle [tex]$\theta$[/tex]:
[tex]\[ \theta = 360^{\circ} - 98^{\circ} \][/tex]
[tex]\[ \theta = 262^{\circ} \][/tex]
So, the angle with the same cosine value as [tex]$98^{\circ}$[/tex] is [tex]\( \boxed{262^{\circ}} \)[/tex].
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Thus, the angle with the same sine value as [tex]$98^{\circ}$[/tex] is [tex]\( 82^{\circ} \)[/tex], and the angle with the same cosine value as [tex]$98^{\circ}$[/tex] is [tex]\( 262^{\circ} \)[/tex].
