Solve for $w$: $4 sin ^2(w)=\cos (w)-1$
Answer (2)
To solve the equation 4 sin 2 ( w ) = cos ( w ) − 1 , we rewrite it using trigonometric identities to form a quadratic equation in terms of cos ( w ) . The only valid solution is cos ( w ) = 1 , leading to the general solution of w = 2 nπ for any integer n .
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Rewrite the equation using the identity sin 2 ( w ) = 1 − cos 2 ( w ) .
Simplify the equation to a quadratic form in terms of cos ( w ) : 4 cos 2 ( w ) + cos ( w ) − 5 = 0 .
Solve the quadratic equation, which gives cos ( w ) = 1 or cos ( w ) = − 4 5 .
Since − 1 ≤ cos ( w ) ≤ 1 , the only valid solution is cos ( w ) = 1 , which means w = 2 nπ , where n is an integer. Therefore, the final answer is 2 nπ .
Explanation
Understanding the Problem We are given the equation 4 sin 2 ( w ) = cos ( w ) − 1 . Our goal is to solve for w . We can use the trigonometric identity sin 2 ( w ) + cos 2 ( w ) = 1 to rewrite the equation in terms of cos ( w ) only.
Rewriting the Equation Using the identity sin 2 ( w ) = 1 − cos 2 ( w ) , we can rewrite the equation as 4 ( 1 − cos 2 ( w )) = cos ( w ) − 1 .
Simplifying to a Quadratic Expanding and rearranging the equation, we get a quadratic equation in terms of cos ( w ) . This results in 4 − 4 cos 2 ( w ) = cos ( w ) − 1 , which simplifies to 4 cos 2 ( w ) + cos ( w ) − 5 = 0 .
Solving the Quadratic Let x = cos ( w ) . The equation becomes 4 x 2 + x − 5 = 0 . We can factor the quadratic as ( 4 x + 5 ) ( x − 1 ) = 0 . Thus, x = 1 or x = − 4 5 .
Considering the Range of Cosine Since x = cos ( w ) , we have cos ( w ) = 1 or cos ( w ) = − 4 5 . However, since − 1 ≤ cos ( w ) ≤ 1 , the only possible solution is cos ( w ) = 1 .
Finding the General Solution Now we solve for w when cos ( w ) = 1 . The general solution is w = 2 nπ , where n is an integer.
Final Answer Therefore, the solution for w is w = 2 nπ , where n is an integer.
Examples
Trigonometric equations like this one are used in many fields of science and engineering, such as physics (wave motion, oscillations), electrical engineering (AC circuits), and computer graphics (rotations, projections). For example, when analyzing the motion of a pendulum, you might encounter an equation involving trigonometric functions that needs to be solved to determine the pendulum's position at a given time. Similarly, in electrical engineering, solving trigonometric equations is crucial for understanding the behavior of alternating current circuits.
