Find the value of [tex]$x$[/tex] if [tex]$\tan 20^{\circ}=\cot (x+30)^{\circ}$[/tex]
Answer (2)
Use the identity cot θ = tan ( 9 0 ∘ − θ ) to rewrite the equation.
Apply the property that if tan a = tan b , then a = b + 180 n for some integer n .
Simplify the equation and solve for x .
Find the value of x when n = 0 : 40 .
Explanation
Understanding the Problem We are given the equation tan 2 0 ∘ = cot ( x + 30 ) ∘ and we need to find the value of x .
Using Trigonometric Identity Recall the identity cot θ = tan ( 9 0 ∘ − θ ) . Using this identity, we can rewrite the given equation as tan 2 0 ∘ = tan ( 9 0 ∘ − ( x + 30 ) ∘ ) .
Applying Tangent Periodicity Since the tangent function has a period of 18 0 ∘ , if tan a = tan b , then a = b + 180 n for some integer n . Therefore, we have 20 = 90 − ( x + 30 ) + 180 n .
Simplifying the Equation Simplifying the equation, we get 20 = 90 − x − 30 + 180 n , which gives 20 = 60 − x + 180 n .
Solving for x Solving for x , we have x = 60 − 20 + 180 n , so x = 40 + 180 n .
Finding the Solution For n = 0 , we have x = 40 . Let's check if x = 40 is a valid solution: cot ( 40 + 30 ) ∘ = cot 7 0 ∘ = tan ( 90 − 70 ) ∘ = tan 2 0 ∘ . Thus, x = 40 is a valid solution.
Final Answer Therefore, the value of x is 4 0 ∘ .
Examples
Imagine you are designing a robotic arm that needs to reach a specific angle. The equation tan 2 0 ∘ = cot ( x + 30 ) ∘ can help you determine the required angle x for one of the arm's joints, given a fixed angle of 2 0 ∘ for another part of the mechanism. This ensures the arm accurately reaches its target position. Trigonometric relationships are crucial in robotics for calculating angles and positions in space, allowing for precise movements and tasks.
The value of x is determined to be 4 0 ∘ using trigonometric identities and properties. By rewriting the cotangent in terms of tangent and applying the periodicity of the tangent function, we find that x = 40 + 180 n , leading to the principal solution of 4 0 ∘ . We confirmed this solution is correct by checking it against the original equation.
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