Solving Trigonometric Equation $\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$ A Step-by-Step Guide

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Hey there, math enthusiasts! Ever stumbled upon a trigonometric equation that looks like it belongs in a super-secret math club? Well, you're not alone! Trigonometric equations can seem daunting at first glance, but with a little bit of trigonometric identity magic and a step-by-step approach, you can conquer even the most complex-looking problems. Today, we're diving deep into a fascinating equation: $\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$. Buckle up, because we're about to embark on a mathematical adventure that will not only solve this specific equation but also equip you with the tools to tackle similar challenges in the future. We will solve this trigonometric equation in the range $0^\circ<x<90^\circ$.

Decoding the Trigonometric Puzzle: Setting the Stage

Before we jump into the nitty-gritty details of solving this equation, let's take a moment to understand the landscape. We're dealing with an equation that involves sines and cosines, which are the fundamental building blocks of trigonometry. These functions relate angles to the ratios of sides in a right triangle, and they dance together in a beautiful, interconnected way. The equation itself presents a ratio of sines and cosines on both sides, suggesting that we might be able to use trigonometric identities to simplify things.

But what are trigonometric identities, you ask? Well, imagine them as the secret weapons in our mathematical arsenal. They are equations that are always true for any value of the angle, and they allow us to rewrite trigonometric expressions in different forms. Think of them as a translator, helping us convert from one trigonometric language to another. Some of the most common and useful identities include:

• Sum-to-product identities: These identities help us express sums or differences of sines and cosines as products.
• Product-to-sum identities: As the name suggests, these identities do the opposite, converting products of sines and cosines into sums or differences.
• Angle addition/subtraction identities: These identities tell us how to expand expressions like $\sin(A + B)$ or $\cos(A - B)$.
• Double-angle identities: These identities provide formulas for $\sin(2A)$, $\cos(2A)$, and $\tan(2A)$.

Our goal is to strategically apply these identities to simplify the given equation and isolate the variable $x$. It's like piecing together a puzzle, where each identity is a piece that helps us reveal the bigger picture.

The Art of Simplification: A Journey Through Trigonometric Identities

Now, let's get our hands dirty and start simplifying the equation. Our initial equation is:

$\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$

The right-hand side of the equation looks like a potential candidate for simplification using product-to-sum identities. Remember, these identities help us convert products of trigonometric functions into sums or differences. Let's focus on the numerator and denominator separately.

Taming the Numerator: $\cos(68^\circ)\sin(24^\circ)$

We can use the following product-to-sum identity:

$\cos(A)\sin(B) = \frac{1}{2}[\sin(A + B) - \sin(A - B)]$

Applying this to our numerator, we get:

$\cos(68^\circ)\sin(24^\circ) = \frac{1}{2}[\sin(68^\circ + 24^\circ) - \sin(68^\circ - 24^\circ)]$

Simplifying the angles, we have:

$\cos(68^\circ)\sin(24^\circ) = \frac{1}{2}[\sin(92^\circ) - \sin(44^\circ)]$

Now, we can use the identity $\sin(180^\circ - \theta) = \sin(\theta)$ to rewrite $\sin(92^\circ)$ as $\sin(88^\circ)$. This gives us:

$\cos(68^\circ)\sin(24^\circ) = \frac{1}{2}[\sin(88^\circ) - \sin(44^\circ)]$

Conquering the Denominator: $\cos(80^\circ)\cos(64^\circ)$

For the denominator, we'll use another product-to-sum identity:

$\cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$

Applying this to our denominator, we get:

$\cos(80^\circ)\cos(64^\circ) = \frac{1}{2}[\cos(80^\circ + 64^\circ) + \cos(80^\circ - 64^\circ)]$

Simplifying the angles, we have:

$\cos(80^\circ)\cos(64^\circ) = \frac{1}{2}[\cos(144^\circ) + \cos(16^\circ)]$

Again, we can use the identity $\cos(180^\circ - \theta) = -\cos(\theta)$ to rewrite $\cos(144^\circ)$ as $-\cos(36^\circ)$. This gives us:

$\cos(80^\circ)\cos(64^\circ) = \frac{1}{2}[-\cos(36^\circ) + \cos(16^\circ)]$

Putting it All Together: The Simplified Equation

Now that we've simplified both the numerator and the denominator, let's plug them back into our original equation. Remember, we had:

$\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$

Substituting our simplified expressions, we get:

$\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\frac{1}{2}[\sin(88^\circ) - \sin(44^\circ)]}{\frac{1}{2}[-\cos(36^\circ) + \cos(16^\circ)]}$

The $\frac{1}{2}$ factors cancel out, leaving us with:

$\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\sin(88^\circ) - \sin(44^\circ)}{-\cos(36^\circ) + \cos(16^\circ)}$

The Sum-to-Product Transformation: Unveiling Hidden Relationships

Our equation is looking cleaner, but we can still do better! Notice that the numerator and denominator both involve differences of trigonometric functions. This is a perfect opportunity to use sum-to-product identities.

Transforming the Numerator: $\sin(88^\circ) - \sin(44^\circ)$

We'll use the following sum-to-product identity:

$\sin(A) - \sin(B) = 2\cos(\frac{A + B}{2})\sin(\frac{A - B}{2})$

Applying this to our numerator, we get:

$\sin(88^\circ) - \sin(44^\circ) = 2\cos(\frac{88^\circ + 44^\circ}{2})\sin(\frac{88^\circ - 44^\circ}{2})$

Simplifying the angles, we have:

$\sin(88^\circ) - \sin(44^\circ) = 2\cos(66^\circ)\sin(22^\circ)$

Transforming the Denominator: $-\cos(36^\circ) + \cos(16^\circ)$

We'll use another sum-to-product identity:

$\cos(A) - \cos(B) = -2\sin(\frac{A + B}{2})\sin(\frac{A - B}{2})$

To match the form of the identity, we rewrite our denominator as $\cos(16^\circ) - \cos(36^\circ)$. Applying the identity, we get:

$\cos(16^\circ) - \cos(36^\circ) = -2\sin(\frac{16^\circ + 36^\circ}{2})\sin(\frac{16^\circ - 36^\circ}{2})$

Simplifying the angles, we have:

$\cos(16^\circ) - \cos(36^\circ) = -2\sin(26^\circ)\sin(-10^\circ)$

Since $\sin(-x) = -\sin(x)$, we can rewrite this as:

$\cos(16^\circ) - \cos(36^\circ) = 2\sin(26^\circ)\sin(10^\circ)$

The Grand Reveal: A Much Simpler Equation

Plugging our transformed numerator and denominator back into the equation, we get:

$\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{2\cos(66^\circ)\sin(22^\circ)}{2\sin(26^\circ)\sin(10^\circ)}$

The 2's cancel out, leaving us with:

$\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\cos(66^\circ)\sin(22^\circ)}{\sin(26^\circ)\sin(10^\circ)}$

The Final Act: Unveiling the Solution for x

We're getting closer to the finish line! Now, let's make a crucial observation. We can rewrite $\cos(66^\circ)$ as $\sin(90^\circ - 66^\circ) = \sin(24^\circ)$ and $\sin(26^\circ)$ as $\cos(90^\circ - 26^\circ) = \cos(64^\circ)$. Substituting these into our equation, we get:

$\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\sin(24^\circ)\sin(22^\circ)}{\cos(64^\circ)\sin(10^\circ)}$

Wait a minute! This looks very close to our original equation. Remember, our original equation was:

$\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$

Let's go back and check our work to see if we can relate $\frac{\sin(22^\circ)}{\sin(10^\circ)}$ to $\frac{\cos(68^\circ)}{\cos(80^\circ)}$. This is a bit tricky, but we will rewrite sines into cosine using the complementary angle identity, which gives $\sin(22^\circ) = \cos(68^\circ)$ and $\sin(10^\circ) = \cos(80^\circ)$. Therefore, the equation becomes $\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\sin(24^\circ)\cos(68^\circ)}{\cos(64^\circ)\cos(80^\circ)}$

Comparing the right hand side of this equation and the original one, they look identical!

Now, let's bring back the original equation: $\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$.

This implies that $\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\sin(24^\circ)\cos(68^\circ)}{\cos(64^\circ)\cos(80^\circ)}$.

Therefore, it seems like the simplification did not lead us anywhere since the equation is now exactly the same as the starting point. Let's rewind a little bit and look for another approach.

Let's go back to the equation $\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\sin(88^\circ) - \sin(44^\circ)}{-\cos(36^\circ) + \cos(16^\circ)}$

Instead of applying sum-to-product to the denominator, let's think about it in a different way. Remember the original equation, we have $\cos(80^\circ)$ and $\cos(64^\circ)$ in the denominator. Can we somehow relate these angles to 36 and 16? Notice that $80 = 96 - 16$ and $64 = 100 - 36$. These does not seem to be any direct relation between these angles.

Another approach we can try is to directly compare the expression. $\frac{\sin(x)}{\cos(8^\circ-x)} = \frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$

$\sin(x)\cos(80^\circ)\cos(64^\circ) = \cos(68^\circ)\sin(24^\circ)\cos(8^\circ-x)$

This looks even more complicated since now we have both $x$ inside and outside of trigonometric functions.

Let's rethink our strategy. We can use a calculator to compute the value of the RHS.

$\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)} \approx \frac{0.3746 \times 0.4067}{0.1736 \times 0.4384} \approx 2.000$

Now, the equation is $\frac{\sin(x)}{\cos(8^\circ-x)} = 2$.

$\sin(x) = 2\cos(8^\circ-x)$ $\sin(x) = 2[\cos(8^\circ)\cos(x) + \sin(8^\circ)\sin(x)]$ $\sin(x) = 2\cos(8^\circ)\cos(x) + 2\sin(8^\circ)\sin(x)$ $\sin(x)(1 - 2\sin(8^\circ)) = 2\cos(8^\circ)\cos(x)$ $\tan(x) = \frac{2\cos(8^\circ)}{1 - 2\sin(8^\circ)} \approx \frac{2 \times 0.9903}{1 - 2 \times 0.1392} \approx \frac{1.9806}{0.7216} \approx 2.7448$

$x = \arctan(2.7448) \approx 70^\circ$

Conclusion: Triumph Over Trigonometry

And there you have it, folks! We've successfully solved the trigonometric equation $\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$ for $0^\circ<x<90^\circ$. It was a journey filled with trigonometric identities, strategic simplifications, and a bit of perseverance. But in the end, we emerged victorious!

The key takeaways from this adventure are:

• Trigonometric identities are your best friends. Master them, and you'll be able to transform even the most intimidating equations.
• Simplification is key. Break down complex expressions into smaller, manageable pieces.
• Don't be afraid to try different approaches. If one path doesn't lead to a solution, explore others.
• Sometimes, a calculator can be a lifesaver. Don't hesitate to use it when appropriate.

Remember, solving trigonometric equations is like learning a new language. It takes practice, patience, and a willingness to experiment. But with each equation you conquer, you'll become more fluent in the language of trigonometry. So, keep exploring, keep practicing, and keep unlocking the mysteries of math!

Keywords Targeted

• Trigonometric Equations: Used extensively throughout the article, focusing on the step-by-step solution of the given equation.
• Trigonometric Identities: Emphasized as the core tool for simplifying and solving trigonometric equations, with explanations of various identities like sum-to-product and product-to-sum.
• Solving Trigonometric Equations: Directly addresses the process and techniques involved in solving such equations, highlighting simplification and strategic approaches.
• Angle Addition/Subtraction Identities: Mentioned and utilized to expand and simplify trigonometric expressions.
• Product-to-Sum Identities: Detailed explanation and application for converting products of trigonometric functions into sums or differences.
• Sum-to-Product Identities: Detailed explanation and application for converting sums or differences of trigonometric functions into products.
• Double-Angle Identities: Mentioned as part of the fundamental trigonometric tools.
• Simplify Trigonometric Expressions: Emphasized as a key step in solving equations, involving the use of identities to make equations more manageable.
• Trigonometry: A broad term covered throughout the article, ensuring relevance to the general topic.
• Solve $\frac{\sin(x)}{\cos(8^\circ-x)}=\frac{\cos(68^\circ)\sin(24^\circ)}{\cos(80^\circ)\cos(64^\circ)}$: The specific equation is mentioned multiple times, ensuring the article targets users searching for its solution.