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Chapter 4: Problem 85

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 1/2 because, in every 30-60-90 degree right triangle, the length of the side opposite the 30-degree angle is half the length of the hypotenuse, and sine is defined as the ratio of the opposite side to the hypotenuse.

Step by step solution

01

Understanding 30-60-90 triangle

First, recognize that in a 30-60-90 degree right triangle, the sides are in the ratio 1:鈭�3:2. This means that the smallest side (across 30掳) is half the length of the hypotenuse.
02

Definition of Sine

Recall that the trigonometric function sine (sin) for any angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
03

Applying Sine to 30 degrees in a right triangle

Now, apply the definition of sine to a 30-degree angle in a right triangle. The side opposite the 30-degree angle is half the length of the hypotenuse (as per the 1:鈭�3:2 ratio), so the sine of 30 degrees, or \(\sin 30^{\circ}\), is \(\frac{Opposite Side}{Hypotenuse} = \frac{1}{2}\)

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