91Ó°ÊÓ

Understanding trigonometric identities is crucial when solving equations involving trigonometric functions. Specifically, tangent and cotangent identities allow us to relate these two functions, which can significantly simplify the process of solving equations. For example, a basic relationship to remember is that \( \cot x = \frac{1}{\tan x} \). This reciprocal identity becomes very useful in converting a cotangent expression into a tangent one, making it easier to handle since we're only dealing with a single trigonometric function. \
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When faced with a trigonometric equation that contains both tangent and cotangent terms, we can use this identity to rewrite the cotangent terms in terms of tangent, reducing the complexity of the equation. As seen in the given exercise, converting \( 5 \cot 3x \) into \( \frac{5}{3 \tan x} \) is an application of this identity, setting the stage for transforming the equation into a quadratic form.

In trigonometry, it's not uncommon to encounter quadratic equations where the variable is a trigonometric function such as \( \tan x \). The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \) , \( b \) , and \( c \) are constants, and \( x \) is the variable we need to solve for. \
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To solve a trigonometric quadratic equation, we may need to manipulate the equation to fit this standard form, just as we do with regular quadratics. This often involves combining terms, factoring, expanding, and using trigonometric identities. As observed in the exercise, after using the identity for cotangent, what initially seemed to be a complicated trigonometric equation involving \( \tan x \) and \( \cot x \) was manipulated into a quadratic equation in terms of \( \tan x \). This opens up the possibility of applying the quadratic formula to find solutions for \( \tan x \) and, by extension, for \( x \).

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a powerful tool for finding the roots of any quadratic equation, provided that \( a eq 0 \). When we have a trigonometric function such as \( \tan x \) in place of \( x \), we apply the formula in the same way, treating \( \tan x \) as the variable. \
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The solutions derived from the quadratic formula give us potential values of \( \tan x \) that we can use to find the corresponding angles \( x \). However, we must remember that not all solutions may be valid for a given trigonometric equation. For instance, when considering the tangent function, we need to ensure that our solutions for \( x \) lie within its principal domain, which is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). If a solution falls outside this range, it's disregarded as it doesn't satisfy the original equation's conditions. This step is crucial and often serves as a final check to ensure the correctness of the solutions obtained.