What is the value of [tex]$\tan \left(60^{\circ}\right)$[/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 (2)
Recall the definition of tangent: tan ( θ ) = c o s ( θ ) s i n ( θ ) .
Remember the values: sin ( 6 0 ∘ ) = 2 3 and cos ( 6 0 ∘ ) = 2 1 .
Calculate tan ( 6 0 ∘ ) as c o s ( 6 0 ∘ ) s i n ( 6 0 ∘ ) = 2 1 2 3 .
Simplify the expression to find the final answer: 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 sine and cosine at this angle to determine the tangent.
Tangent Definition Recall that the tangent of an angle is defined as the ratio of the sine to the cosine of that angle: tan ( θ ) = cos ( θ ) sin ( θ )
Sine and Cosine Values We know that sin ( 6 0 ∘ ) = 2 3 and cos ( 6 0 ∘ ) = 2 1 .
Calculate Tangent Now, we can find the tangent of 6 0 ∘ by dividing sin ( 6 0 ∘ ) by cos ( 6 0 ∘ ) :
tan ( 6 0 ∘ ) = cos ( 6 0 ∘ ) sin ( 6 0 ∘ ) = 2 1 2 3 = 2 3 × 1 2 = 3
Final Answer Therefore, the value of tan ( 6 0 ∘ ) is 3 .
Examples
Understanding trigonometric values like tan ( 6 0 ∘ ) = 3 is crucial in various real-world applications, such as calculating the height of a tree or building using angles of elevation and distances. For instance, if you stand a certain distance away from a tree and measure the angle of elevation to the top of the tree to be 6 0 ∘ , knowing the distance to the tree allows you to calculate the tree's height using the tangent function. If you are 10 meters away from the base of the tree, the height of the tree would be 10 × tan ( 6 0 ∘ ) = 10 3 meters.
The value of tan ( 6 0 ∘ ) is 3 . Therefore, the correct answer is option B: 3 .
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