Sin(30°)

Definition

sin(30°) is the sine of a 30-degree angle, which represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. In the context of the unit circle, this value corresponds to a specific point on the circle, highlighting its relationship with trigonometric functions and geometric interpretations.

5 Must Know Facts For Your Next Test

1. sin(30°) equals 0.5, meaning that in a right triangle with a 30-degree angle, the length of the opposite side is half that of the hypotenuse.
2. On the unit circle, sin(30°) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle, specifically (√3/2, 1/2).
3. The sine function is periodic with a period of 360°, which means that sin(30°) will also equal sin(390°) and sin(-330°).
4. sin(30°) is part of common angles in trigonometry, along with sin(0°), sin(45°), sin(60°), and sin(90°), forming a basis for calculating other sine values.
5. The value sin(30°) is often used in problems involving special triangles, such as the 30-60-90 triangle, where the side lengths follow a specific ratio.

Review Questions

• How can you demonstrate that sin(30°) equals 0.5 using a right triangle?
  • To show that sin(30°) equals 0.5 using a right triangle, consider a 30-60-90 triangle. In this type of triangle, if we set the hypotenuse to 1 unit, then by the properties of this triangle, the side opposite the 30-degree angle will be 0.5 units long. The sine function is defined as the ratio of the length of the opposite side to that of the hypotenuse. Therefore, sin(30°) = opposite/hypotenuse = 0.5/1 = 0.5.
• Explain how sin(30°) relates to coordinates on the unit circle.
  • In the unit circle, angles are measured from the positive x-axis, and each angle corresponds to a point on the circle with coordinates (cos(θ), sin(θ)). For an angle of 30°, this translates to coordinates (√3/2, 1/2). The y-coordinate gives us sin(30°), which is 1/2. This relationship illustrates how trigonometric functions can be visualized geometrically on a coordinate plane.
• Evaluate how knowing that sin(30°) equals 0.5 can help in solving more complex trigonometric equations or identities.
  • Knowing that sin(30°) equals 0.5 is fundamental for solving complex trigonometric equations or identities because it serves as a reference point in calculations involving other angles and functions. For instance, it can simplify expressions like sin(150°), which can be evaluated as sin(180° - 30°). Additionally, this knowledge helps in deriving other sine values through identities such as angle addition or subtraction formulas, enabling easier computation and deeper understanding in more advanced trigonometric contexts.

Related terms

Unit Circle: A circle with a radius of one, centered at the origin of a coordinate system, where the coordinates of any point on the circle correspond to the cosine and sine values of an angle.

Right Triangle: A triangle that has one angle measuring 90 degrees, where sine, cosine, and tangent functions can be easily defined in relation to its sides.

Trigonometric Ratios: Ratios that relate the angles of a triangle to the lengths of its sides, including sine, cosine, and tangent.