To find the expression for the hypotenuse \( c \) of the right triangle with one acute angle of 45 degrees, we can use the relationships between the sides of a right triangle and the angles.
In a right triangle with a 45-degree angle, the following relationships hold:
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The side opposite the 45-degree angle (which is length \( a \)) is equal to the side adjacent to the 45-degree angle (which is length \( b \)) because in a 45-45-90 triangle, both legs are of equal length.
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The relationships involving trigonometric functions are as follows:
- \(\tan(45^\circ) = 1\) which means \( a = b \).
- \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\) and \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\).
From the right triangle's definition and the properties mentioned, we know that:
\[ c = \frac{a}{\sin(45^\circ)} \quad \text{and} \quad c = \frac{b}{\cos(45^\circ)} \]
Substituting \(\sin(45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}}\):
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From \( c = \frac{a}{\sin(45^\circ)} \): \[ c = a \cdot \sqrt{2} \]
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From \( c = \frac{b}{\cos(45^\circ)} \): \[ c = b \cdot \sqrt{2} \]
Therefore, either \( c = a \sqrt{2} \) or \( c = b \sqrt{2} \).
Looking at the provided options:
- \( b \over \tan 45^\circ \) simplifies to \( b \) (which doesn't equal \( c \)),
- \( b \over \cos 45^\circ = b \cdot \sqrt{2} = c\),
- \( a \cos 45^\circ \) simplifies to \( 0.7071a \) (which doesn't equal \( c \)),
- \( a \tan 45^\circ \) simplifies to \( a \) (which doesn't equal \( c \)).
Thus, the correct expression that shows the value of \( c \) is:
\[ \frac{b}{\cos 45^\circ} \]
1 answer