Tan(60°)

Definition

tan(60°) refers to the tangent of a 60-degree angle, which in trigonometry is a ratio of the length of the opposite side to the length of the adjacent side in a right triangle. In the context of the unit circle, tan(60°) can also be understood as the y-coordinate divided by the x-coordinate of the point on the unit circle corresponding to 60 degrees. This connection illustrates how angles relate to coordinate geometry and trigonometric values.

5 Must Know Facts For Your Next Test

1. The exact value of tan(60°) is √3, or approximately 1.732.
2. In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2, which helps derive tan(60°).
3. Tangent is periodic with a period of π, meaning tan(60°) also equals tan(60° + nπ) for any integer n.
4. On the unit circle, the coordinates for 60 degrees are (1/2, √3/2), leading to tan(60°) being calculated as (√3/2) / (1/2) = √3.
5. Understanding tan(60°) is crucial for solving problems involving right triangles and circular functions.

Review Questions

• How can you derive the value of tan(60°) using properties of special triangles?
  • To derive tan(60°), you can use a 30-60-90 triangle. In this type of triangle, if you consider an angle of 60 degrees, the opposite side (which corresponds to tan) is √3 times longer than the adjacent side, which measures 1. Therefore, tan(60°) = opposite/adjacent = √3/1 = √3. This relationship highlights how specific angle measures relate to exact side ratios in special right triangles.
• Discuss how tan(60°) relates to points on the unit circle and what that means geometrically.
  • On the unit circle, an angle of 60 degrees corresponds to the point (1/2, √3/2). Tangent is defined as the ratio of y to x coordinates at that point. Therefore, tan(60°) can be calculated as (√3/2) / (1/2), which simplifies to √3. Geometrically, this demonstrates how tangent represents slope or steepness when viewing points on the circle, connecting circular motion with linear functions.
• Evaluate how knowing tan(60°) can help solve more complex trigonometric problems involving angles and their properties.
  • Knowing that tan(60°) equals √3 allows you to simplify and solve more complex problems involving trigonometric identities and equations. For instance, if you're working with angles related through addition or subtraction formulas, such as tan(45° + 15°), recognizing that 15 degrees can be expressed using 30 and 60-degree relationships enables you to apply known values like tan(60°) directly. This ability to connect various angles with known values enhances problem-solving efficiency and reinforces understanding of trigonometric functions in various contexts.

Related terms

Sine: Sine is a trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.

Cosine: Cosine is a trigonometric function defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

Unit Circle:

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions for all angles.