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Chapter 4: Problem 78

In right triangle trigonometry, explain why \(\sin 30^{\circ}=\frac{1}{2}\) regardless of the size of the triangle.

Short Answer

Expert verified
The sine of 30 degrees is always \(\frac{1}{2}\), regardless of the size of the triangle, because of the special proportions in a 30-60-90 right triangle. In such a triangle, the side opposite the 30 degree angle is always half the length of the hypotenuse, so their ratio (which is the sine of 30 degrees) is always \(\frac{1}{2}\).

Step by step solution

01

Understand what sine represents

In a right triangle, the sine of an angle (abbreviated as 'sin') is the ratio of the length of the side opposite of the angle to the length of the hypotenuse (which is the largest side, opposite to the right angle). In mathematical terms, for an angle \(A\), \(\sin A = \frac{Opposite}{Hypotenuse}\).
02

Recognize the special properties of a 30-60-90 triangle

A 30-60-90 triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of a 30-60-90 triangle always have the same proportions: if the shortest side (opposite the 30 degree angle) has length \(x\), then the longer leg (opposite the 60 degree angle) has length \(x\sqrt{3}\), and the hypotenuse (opposite the 90 degree angle) has length \(2x\). This is always true, no matter the size of the triangle, because it's a consequence of the triangle's angles. Whenever you have a 30 degree angle in a right triangle, the side opposite that angle is half the length of the hypotenuse.
03

Apply the definition of sine to a 30-60-90 triangle

If we apply the definition of sine to a 30 degree angle in a 30-60-90 triangle, we get: \(\sin 30^{\circ} = \frac{Opposite}{Hypotenuse} = \frac{x}{2x} = \frac{1}{2}\). No matter the size of the triangle (no matter the specific value of 'x'), this ratio will always simplify to \(\frac{1}{2}\), because the side lengths are always in the same proportion. This is why the sine of 30 degrees is always \(\frac{1}{2}\), regardless of the size of the triangle.

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