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 range of 0 to 2 π , which is 6 π .
Determine the reference angle for 6 π , which is 6 π itself since it's in the first quadrant.
Calculate the cotangent of the reference angle: cot ( 6 π ) = 3 .
Since 6 13 π is in the first quadrant where cotangent is positive, the final answer is 3 .
Explanation
Problem Analysis We are asked to find the exact value of cot ( 6 13 π ) using reference angles. Let's break this down step by step.
Finding a Coterminal Angle First, we need to find an angle that is coterminal with 6 13 π and lies between 0 and 2 π . To do this, we can subtract multiples of 2 π from 6 13 π until we get an angle in the desired range. Since 2 π = 6 12 π , we can subtract 6 12 π from 6 13 π to get 6 13 π − 6 12 π = 6 π . So, 6 π is a coterminal angle to 6 13 π .
Determining the Reference Angle Now we need to find the reference angle for 6 π . Since 6 π is already in the first quadrant, its reference angle is simply itself. Therefore, the reference angle is 6 π .
Calculating the Cotangent of the Reference Angle Next, we need to find the value of the cotangent function for the reference angle 6 π . Recall that cot ( θ ) = s i n ( θ ) c o s ( θ ) . So, cot ( 6 π ) = s i n ( 6 π ) c o s ( 6 π ) . We know that cos ( 6 π ) = 2 3 and sin ( 6 π ) = 2 1 . Therefore, cot ( 6 π ) = 2 1 2 3 = 3 .
Determining the Sign of the Cotangent Now we need to determine the sign of the cotangent function in the quadrant in which the original angle 6 13 π lies. Since 6 13 π is coterminal with 6 π , it lies in the first quadrant. In the first quadrant, all trigonometric functions are positive, so the cotangent is positive.
Finding the Exact Value Finally, we apply the appropriate sign to the cotangent value of the reference angle to find the exact value of cot ( 6 13 π ) . Since the cotangent is positive in the first quadrant, we have cot ( 6 13 π ) = cot ( 6 π ) = 3 .
Final Answer Therefore, the exact value of cot ( 6 13 π ) is 3 .
Examples
Understanding reference angles and trigonometric functions like cotangent is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, you often need to calculate trigonometric values for angles outside the standard 0 to 2 π range. Using reference angles simplifies these calculations and allows for accurate modeling and prediction of system behavior.
To find cot 6 13 π , we determined that a coterminal angle is 6 π . The reference angle for this is also 6 π , and we calculated cot 6 π = 3 . Therefore, the exact value of cot 6 13 π is 3 .
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