Given: [tex]\tan A=-\sqrt{15}[/tex]
What is the value of [tex]\tan \left(A-\frac{\pi}{4}\right)[/tex] ?
A. [tex]\frac{\sqrt{15}+1}{1-\sqrt{15}}[/tex]
B. [tex]\frac{-\sqrt{15}+1}{1+\sqrt{15}}[/tex]
C. [tex]\frac{\sqrt{15}-1}{1+\sqrt{15}}[/tex]
D. [tex]\frac{-\sqrt{15}-1}{1-\sqrt{15}}[/tex]
Answer (2)
Apply the tangent subtraction formula: tan ( A − B ) = 1 + t a n A t a n B t a n A − t a n B .
Substitute B = 4 π , so tan B = 1 .
Substitute tan A = − 15 and tan ( 4 π ) = 1 into the formula.
Simplify the expression to get the final answer: 1 − 15 − 15 − 1 .
Explanation
Problem Setup We are given that tan A = − 15 , and we want to find the value of tan ( A − 4 π ) .
Tangent Subtraction Formula We will use the tangent subtraction formula, which states that tan ( A − B ) = 1 + t a n A t a n B t a n A − t a n B . In our case, B = 4 π , so tan B = tan ( 4 π ) = 1 .
Substitution Substituting the given values into the formula, we have tan ( A − 4 π ) = 1 + tan A tan ( 4 π ) tan A − tan ( 4 π ) = 1 + ( − 15 ) ( 1 ) − 15 − 1 = 1 − 15 − 15 − 1 .
Final Answer Thus, the value of tan ( A − 4 π ) is 1 − 15 − 15 − 1 .
Examples
Imagine you're designing a robotic arm that needs to operate at a specific angle, but there's a slight adjustment needed due to the environment. Using trigonometric identities like the tangent subtraction formula allows you to calculate the new angle accurately, ensuring the arm performs its task correctly. This is crucial in manufacturing, surgery, and even space exploration where precision is key. By understanding these formulas, engineers can make precise adjustments to angles in various mechanical and electronic systems, optimizing performance and safety.
We used the tangent subtraction formula to find that tan ( A − 4 π ) = 1 − 15 − 15 − 1 . The correct choice from the provided options is D. This approach illustrates how trigonometric identities can help with angle calculations effectively.
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