What is the exact value of $\cot \left(\frac{3 \pi}{4}\right) $?
Answer (2)
The problem asks for the exact value of cot ( 4 3 π ) .
Recall that cot ( x ) = s i n ( x ) c o s ( x ) .
Determine that cos ( 4 3 π ) = − 2 2 and sin ( 4 3 π ) = 2 2 .
Calculate cot ( 4 3 π ) = 2 2 − 2 2 = − 1 , so the final answer is − 1 .
Explanation
Understanding the Problem We are asked to find the exact value of cot ( 4 3 π ) . Let's recall the definition of the cotangent function: cot ( x ) = s i n ( x ) c o s ( x ) . Thus, we need to find the values of cos ( 4 3 π ) and sin ( 4 3 π ) .
Finding Cosine and Sine The angle 4 3 π is in the second quadrant. In the second quadrant, the cosine is negative and the sine is positive. We can express 4 3 π as π − 4 π . Therefore, we have: cos ( 4 3 π ) = cos ( π − 4 π ) = − cos ( 4 π ) = − 2 2 sin ( 4 3 π ) = sin ( π − 4 π ) = sin ( 4 π ) = 2 2
Calculating Cotangent Now we can calculate the cotangent: cot ( 4 3 π ) = sin ( 4 3 π ) cos ( 4 3 π ) = 2 2 − 2 2 = − 1
Final Answer Therefore, the exact value of cot ( 4 3 π ) is − 1 .
Examples
Cotangent functions are used in various fields like physics and engineering to describe angles and slopes in periodic phenomena, such as the motion of a pendulum or the behavior of alternating current in electrical circuits. For instance, when analyzing the stability of a structure, engineers use trigonometric functions like cotangent to calculate angles and forces, ensuring the structure remains balanced and secure. Understanding cotangent helps in predicting and controlling these behaviors, making it a fundamental tool in these disciplines.
The exact value of cot ( 4 3 π ) is − 1 . This is calculated using the definitions of cosine and sine for the angle in the second quadrant. Thus, you get cot ( 4 3 π ) = s i n ( 4 3 π ) c o s ( 4 3 π ) = − 1 .
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