What is the complement of [tex]$\cot 30^{\circ}$[/tex]?
Answer (2)
Determine the complementary angle: 9 0 ∘ − 3 0 ∘ = 6 0 ∘ .
Recognize that the complement of cot 3 0 ∘ is tan 6 0 ∘ .
Calculate tan 6 0 ∘ using the values of sin 6 0 ∘ and cos 6 0 ∘ .
State the final answer: 3 .
Explanation
Understanding the problem The problem asks us to find the complement of cot 3 0 ∘ . Let's break this down step by step.
Finding the complementary angle First, recall that the complement of an angle x is 9 0 ∘ − x . In our case, x = 3 0 ∘ , so the complementary angle is 9 0 ∘ − 3 0 ∘ = 6 0 ∘ .
Using the cofunction Next, remember that the complement of a trigonometric function is its cofunction evaluated at the complementary angle. Since we are given cot 3 0 ∘ , its complement is tan ( 9 0 ∘ − 3 0 ∘ ) = tan 6 0 ∘ .
Calculating tan 60 degrees Now, we need to find the value of tan 6 0 ∘ . Recall that 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 . Therefore, tan 6 0 ∘ = 2 1 2 3 = 3 .
Final Answer Thus, the complement of cot 3 0 ∘ is 3 .
Examples
Imagine you're building a ramp for skateboarding. If the angle of the ramp is 30 degrees, you might need to calculate the tangent of the complementary angle (60 degrees) to determine the ratio of the ramp's height to its horizontal length. This ensures the ramp has the desired steepness for performing tricks safely and effectively. In this case, knowing that the tangent of 60 degrees is 3 helps you design the ramp with the correct proportions.
The complement of cot 3 0 ∘ is found by taking the tangent of its complementary angle, which is tan 6 0 ∘ . Calculating tan 6 0 ∘ gives us 3 . Therefore, the answer is 3 .
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