The length [tex]BC[/tex] is [tex]\boxed{14.5\:\text{cm}}[/tex].
Explanation:
The given triangle is a right angle triangle at angle [tex]C[/tex] and the angle [tex]A[/tex] equals [tex]65^{\circ}[/tex].
In a right angled triangle, the two sides which make the right angle or which are perpendicular, are called the base and the height of the right angle triangle. The third side, the largest of the three is called the hypotenuse of the right angled triangle.
From the figure, it is observed that [tex]AC[/tex] is the base, [tex]BC[/tex] is the height and [tex]AB[/tex] is the hypotenuse.
The formula for trigonometric identities [tex]\sin(\theta)[/tex], [tex]\cos(\theta)[/tex] and [tex]\tan(\theta)[/tex], where [tex]\theta[/tex] is the angle between hypotenuse and base, is as shown below.
[tex]\sin\theta =\dfrac{\text{Height}}{\text{Hypotenuse}}\\\cos\theta=\dfrac{\text{Base}}{\text{Hypotenuse}}\\\tan\theta=\dfrac{\text{Height}}{\text{Base}}[/tex]
Substitute [tex]65^{\circ}[/tex] for [tex]\theta[/tex], [tex]16[/tex] for hypotenuse and [tex]x[/tex] for height in the formula for sine of angle [tex]\theta[/tex].
[tex]\boxed{\begin{aligned}\sin(65^{\circ} )& = \dfrac{x}{16}\\0.906307787 \cdot 16& = x\\x&=14.5\end{aligned}}[/tex]
Thus, the length of the side [tex]BC[/tex] is [tex]\boxed{14.5\:\text{cm}}[/tex].
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Answer Details:
Grade: Middle School
Subject: Mathematics
Chapter: Trigonometry
Keywords: trigonometry, right angle, base, height, hypotenuse, sine, triangle, [tex]\theta[/tex], length, side, angles.
