To find the altitude \( h \) of triangle ABC, where \( a = 36 \) cm and \( \angle C = 23^\circ \), we can use the formula for the area of a triangle, which can also be expressed in terms of the base and height:
\[ \text{Area} = \frac{1}{2} \times a \times h \]
We can also use the formula for the area in terms of two sides and the included angle:
\[ \text{Area} = \frac{1}{2} \times b \times c \times \sin(A) \]
In this case, since we only have one side (a) and the angle C, we can modify our method. The side \( a \) will serve as the base (let's say \( b \)).
To find the altitude \( h \) for side \( a \), we can use the following formula relating the altitude and side \( a \):
\[ h = a \cdot \sin(C) \]
- First, we need to compute \( \sin(23^\circ) \):
Using a calculator, \[ \sin(23^\circ) \approx 0.3907 \]
- Now plug this value into the formula for \( h \):
\[ h = 36 , \text{cm} \cdot \sin(23^\circ) \approx 36 , \text{cm} \cdot 0.3907 \approx 14.0852 , \text{cm} \]
- Finally, rounding \( h \) to the nearest whole number:
\[ h \approx 14 , \text{cm} \]
Thus, the length of the altitude \( h \) is approximately 14 cm.
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