To find the value of [tex]\(\sin 60^{\circ}\)[/tex], let's use our knowledge of trigonometry, specifically the values of sine for common angles.
The sine of an angle in a right triangle is found by taking the ratio of the length of the side opposite the angle to the hypotenuse of the triangle. For exact values of sine, cosine, and tangent, it is helpful to remember the values for key angles: 0°, 30°, 45°, 60°, and 90°.
For [tex]\(\sin 60^{\circ}\)[/tex], recall that the sine of 60 degrees is one of the standard trigonometric values. Let's break it down step by step, using geometric reasoning.
First, consider an equilateral triangle where all sides are of equal length and each angle is 60 degrees. By cutting the equilateral triangle in half, we create two 30-60-90 right triangles.
The properties of a 30-60-90 triangle are well known:
- The side opposite the 30° angle is [tex]\( \frac{1}{2} \)[/tex] the hypotenuse.
- The side opposite the 60° angle is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] the hypotenuse.
- The hypotenuse is the longest side.
Now, let’s calculate [tex]\(\sin 60^{\circ}\)[/tex] using this triangle:
- In a 30-60-90 triangle, if the hypotenuse is of length [tex]\(2\)[/tex], then:
- The length of the side opposite the 30° angle (i.e., half the length of the hypotenuse) is [tex]\(1\)[/tex].
- The length of the side opposite the 60° angle is [tex]\( \sqrt{3} \)[/tex].
The sine function is defined as:
[tex]\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
For [tex]\(\theta = 60^{\circ}\)[/tex]:
[tex]\[
\sin 60^{\circ} = \frac{\text{side opposite 60°}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Thus, [tex]\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)[/tex].
