What is tan 30°? 60° 2 1 90° 30° ?3 A. 1 B. ?3 C. \frac{1}{\sqrt{3}} D. \frac{\sqrt{3}}{2} E. ?2 F. \frac{2}{\sqrt{3}}
Question 10 of 10 2 Points What is tan 30°? 60° 2 1 90° 30° ?3 A. 1 B. ?3 C. \frac{1}{\sqrt{3}} D. \frac{\sqrt{3}}{2} E. ?2 F. \frac{2}{\sqrt{3}}
The Correct Answer and Explanation is:
The correct answer is C. 13\frac{1}{\sqrt{3}}.
Explanation:
In trigonometry, the tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This fundamental concept is widely used in solving various problems involving right triangles.
For a 30-60-90 right triangle, the sides are always in a fixed ratio: 1:3\sqrt{3}:2. This ratio originates from an equilateral triangle divided into two smaller right triangles.
To find tan 30°, use the tangent function definition:
tan30∘=oppositeadjacent\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}}
From the triangle, the side opposite 30° is 1, while the adjacent side is 3\sqrt{3}.
Substituting these values:
tan30∘=13\tan 30^\circ = \frac{1}{\sqrt{3}}
This result is significant because 30-60-90 triangles occur frequently in geometry and physics. Recognizing this ratio allows for quick calculations without relying on calculators.
Additionally, this value helps in trigonometric identities and real-world applications such as calculating slopes, angles in projectile motion, and designing structures. Since tan30∘\tan 30^\circ is involved in many formulas, memorizing its value is beneficial for solving advanced problems efficiently.
Understanding these principles strengthens mathematical intuition and problem-solving skills, which are essential in physics, engineering, and other STEM fields. If needed, rationalizing 13\frac{1}{\sqrt{3}} by multiplying the numerator and denominator by 3\sqrt{3} results in 33\frac{\sqrt{3}}{3}, another commonly used equivalent form.
