What is the value of [tex]\tan (60^{\circ})[/tex]?
A. [tex]\frac{1}{2}[/tex]
B. [tex]\sqrt{3}[/tex]
C. [tex]\frac{\sqrt{3}}{2}[/tex]
D. [tex]\frac{1}{\sqrt{3}}[/tex]
Answer (1)
Recall that tan ( θ ) = c o s ( θ ) s i n ( θ ) .
Substitute sin ( 6 0 ∘ ) = 2 3 and cos ( 6 0 ∘ ) = 2 1 into the formula.
Calculate the value: tan ( 6 0 ∘ ) = 2 1 2 3 = 3 .
The final answer is 3 .
Explanation
Problem Analysis The problem asks us to find the value of the tangent function at 6 0 ∘ . We need to recall the values of trigonometric functions for special angles.
Recall Trigonometric Values Recall that tan ( θ ) = c o s ( θ ) s i n ( θ ) . Therefore, tan ( 6 0 ∘ ) = c o s ( 6 0 ∘ ) s i n ( 6 0 ∘ ) . We know that sin ( 6 0 ∘ ) = 2 3 and cos ( 6 0 ∘ ) = 2 1 .
Calculate Tangent Substitute these values into the expression for tan ( 6 0 ∘ ) : tan ( 6 0 ∘ ) = 2 1 2 3 = 2 3 ⋅ 1 2 = 3 Alternatively, we can recall directly that tan ( 6 0 ∘ ) = 3 .
Final Answer The value of tan ( 6 0 ∘ ) is 3 .
Examples
Understanding trigonometric values like tan ( 6 0 ∘ ) is crucial in various fields, such as physics and engineering. For instance, when analyzing the trajectory of a projectile launched at an angle of 6 0 ∘ , the tangent of this angle helps determine the ratio of the vertical and horizontal components of the projectile's initial velocity. This allows engineers to accurately predict the range and height of the projectile, ensuring it hits its intended target. Trigonometry, therefore, serves as a foundational tool in real-world applications involving angles and motion.
