Tan(30°)

Definition

The value of tan(30°) is a trigonometric function that represents the tangent of a 30-degree angle, which equals $$\frac{1}{\sqrt{3}}$$ or approximately 0.577. This value can be derived from the properties of a 30-60-90 triangle, where the tangent function is defined as the ratio of the opposite side to the adjacent side. Understanding this function is crucial as it connects angles to ratios in right triangles and lays the groundwork for further explorations in trigonometry.

5 Must Know Facts For Your Next Test

1. In a 30-60-90 triangle, the sides are in the ratio 1 : $$\sqrt{3}$$ : 2, making it easy to find tan(30°) as $$\frac{1}{\sqrt{3}}$$.
2. The value of tan(30°) can also be expressed as $$\sqrt{3}/3$$, which is an alternative form commonly used in calculations.
3. Tangent is periodic, and tan(30°) also corresponds to other angles: tan(30° + 180°n) for any integer n will yield the same value.
4. Knowing tan(30°) helps in solving problems involving right triangles and is essential for more complex trigonometric equations.
5. tan(30°) is used in various real-world applications, such as calculating slopes, angles in architecture, and engineering.

Review Questions

• How does tan(30°) relate to the properties of a 30-60-90 triangle?
  • Tan(30°) can be directly derived from a 30-60-90 triangle, where the sides are proportionally related. In such a triangle, if we take the side opposite the 30° angle as 1 and the hypotenuse as 2, then by definition, tan(30°) equals the opposite side divided by the adjacent side. The adjacent side would then be $$\sqrt{3}$$, leading to the ratio $$\frac{1}{\sqrt{3}}$$.
• What role does tan(30°) play in understanding the unit circle and its application in trigonometry?
  • In the unit circle, angles are represented in terms of their coordinates on the circle. For an angle of 30°, its coordinates are (cos(30°), sin(30°)), which are related to tangent through tan(30°) = sin(30°)/cos(30°). This relationship helps visualize how tangent values correspond to specific angles on the unit circle, reinforcing concepts such as periodicity and symmetry in trigonometric functions.
• Evaluate how understanding tan(30°) enhances problem-solving skills in real-world contexts involving angles and slopes.
  • Understanding tan(30°) provides a foundational knowledge that can be applied to various real-world scenarios, such as calculating inclines or determining angles of elevation. For example, if an engineer needs to create a ramp with a slope corresponding to an angle of 30°, knowing that this slope translates to a tangent value of $$\frac{1}{\sqrt{3}}$$ aids in precise construction measurements. This concept applies not only to construction but also to physics problems involving forces at angles and navigation where accurate angle determination is crucial.

Related terms

Sine: A trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.

Cosine: A trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

Unit Circle:

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