Use reference angles to find the exact value of [tex]$\cot \frac{13 \pi}{6}$[/tex]
Answer (2)
Find a coterminal angle to 6 13 π within the interval [ 0 , 2 π ) , which is 6 π .
Recall that cot ( θ ) = s i n ( θ ) c o s ( θ ) .
Evaluate cos ( 6 π ) = 2 3 and sin ( 6 π ) = 2 1 .
Calculate cot ( 6 13 π ) = cot ( 6 π ) = 2 1 2 3 = 3 .
Explanation
Finding a Coterminal Angle We are asked to find the exact value of cot ( 6 13 π ) using reference angles. Let's break this down step by step. First, we need to find a coterminal angle to 6 13 π that lies within the interval [ 0 , 2 π ) .
Determining the Coterminal Angle To find a coterminal angle, we can subtract multiples of 2 π from 6 13 π until we get an angle in the desired interval. Since 2 π = 6 12 π , we have 6 13 π − 6 12 π = 6 π So, 6 π is a coterminal angle to 6 13 π . Therefore, cot ( 6 13 π ) = cot ( 6 π ) .
Using Trigonometric Values Now we need to find the value of cot ( 6 π ) . Recall that cot ( θ ) = s i n ( θ ) c o s ( θ ) . We know that cos ( 6 π ) = 2 3 and sin ( 6 π ) = 2 1 .
Calculating the Cotangent Therefore, cot ( 6 π ) = sin ( 6 π ) cos ( 6 π ) = 2 1 2 3 = 3 Thus, cot ( 6 13 π ) = 3 .
Final Answer The exact value of cot ( 6 13 π ) is 3 .
Examples
Imagine you're designing a robotic arm that needs to operate at a specific angle. Knowing the exact cotangent value of angles like 6 13 π helps in programming the arm's movements with precision. This ensures the arm accurately reaches its target, which is crucial in manufacturing, surgery, or even space exploration where accuracy is paramount. Understanding trigonometric functions and reference angles allows for precise control and coordination in various engineering applications.
The exact value of cot 6 13 π is 3 after finding its coterminal angle 6 π and using the values of cosine and sine. The steps include finding the coterminal angle, determining trigonometric values, and calculating the cotangent. Thus, cot 6 13 π = 3 .
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