Factor the trigonometric expression.
[tex]16 \tan ^4 w-24 \tan ^2 w+9[/tex]
Answer (2)
The expression 16 tan 4 w − 24 tan 2 w + 9 can be factored as ( 4 tan 2 w − 3 ) 2 . This is achieved by recognizing it as a quadratic in terms of tan 2 w and using the perfect square trinomial pattern. The final answer is therefore ( 4 tan 2 w − 3 ) 2 .
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Recognize the expression as a quadratic in tan 2 w .
Rewrite the expression as ( 4 tan 2 w ) 2 − 2 ( 4 tan 2 w ) ( 3 ) + 3 2 .
Factor the quadratic expression as a perfect square: ( 4 tan 2 w − 3 ) 2 .
The factored form is ( 4 tan 2 w − 3 ) 2 .
Explanation
Analyzing the Expression Let's analyze the given expression: 16 \t 4 w − 24 \t 2 w + 9 . We need to factor this trigonometric expression. Notice that this expression looks like a quadratic expression in terms of tan 2 w .
Recognizing the Pattern We can rewrite the expression as ( 4 tan 2 w ) 2 − 2 ( 4 tan 2 w ) ( 3 ) + 3 2 . This is a perfect square trinomial of the form a 2 − 2 ab + b 2 , where a = 4 tan 2 w and b = 3 .
Factoring the Expression Now, we can factor the expression as a perfect square: ( 4 tan 2 w − 3 ) 2 . This is the factored form of the given trigonometric expression.
Final Factored Form Therefore, the factored form of 16 tan 4 w − 24 tan 2 w + 9 is ( 4 tan 2 w − 3 ) 2 .
Examples
Factoring trigonometric expressions is useful in simplifying complex equations in physics, such as those found in wave mechanics or optics. For example, when analyzing the interference patterns of light, simplifying trigonometric expressions can help in determining the conditions for constructive or destructive interference. Similarly, in electrical engineering, simplifying such expressions can aid in analyzing AC circuits and signal processing.
