Solving Trigonometric Equations Find The Value Of X

In the realm of mathematics, trigonometric equations hold a significant place, often posing intriguing challenges that require a blend of algebraic manipulation and trigonometric identities to solve. One such equation is presented as follows:

$\sin (4 x-10)^{\circ}=\cos (40-x)^{\circ}$

Our mission is to unravel this equation and determine the value of $x$ that satisfies it. To embark on this mathematical journey, we will leverage the fundamental trigonometric identity that connects sine and cosine functions. This identity states that $\sin(\theta) = \cos(90^{\circ} - \theta)$. By applying this identity, we can transform the given equation into a more manageable form, paving the way for a straightforward solution.

Utilizing Trigonometric Identities

The cornerstone of our approach lies in the astute application of the trigonometric identity $\sin(\theta) = \cos(90^{\circ} - \theta)$. This identity unveils a profound relationship between sine and cosine functions, allowing us to express one in terms of the other. In essence, the sine of an angle is equal to the cosine of its complement, and vice versa. This identity is particularly useful when dealing with equations involving both sine and cosine functions, as it enables us to express them in a unified form.

In our given equation, $\sin (4 x-10)^{\circ}=\cos (40-x)^{\circ}$, we can apply this identity to rewrite the sine term as a cosine term. Specifically, we can express $\sin (4 x-10)^{\circ}$ as $\cos (90^{\circ} - (4x - 10)^{\circ})$. This transformation allows us to rewrite the original equation in terms of cosine functions only, making it easier to manipulate and solve.

Transforming the Equation

By applying the trigonometric identity $\sin(\theta) = \cos(90^{\circ} - \theta)$, we can rewrite the given equation $\sin (4 x-10)^{\circ}=\cos (40-x)^{\circ}$ as follows:

$\cos (90^{\circ} - (4x - 10)^{\circ}) = \cos (40-x)^{\circ}$

This transformation is a pivotal step in our solution process. By expressing both sides of the equation in terms of the same trigonometric function, cosine, we can now equate the arguments of the cosine functions. In other words, if the cosine of two angles is equal, then the angles themselves must be equal (or differ by a multiple of $360^{\circ}$, which we will address later). This simplification allows us to eliminate the trigonometric functions and focus on solving a purely algebraic equation.

Solving for x

Now that we have transformed the equation into $\cos (90^{\circ} - (4x - 10)^{\circ}) = \cos (40-x)^{\circ}$, we can equate the arguments of the cosine functions:

$90^{\circ} - (4x - 10)^{\circ} = (40-x)^{\circ}$

This equation is a linear equation in $x$, which we can solve using standard algebraic techniques. First, we simplify the equation by distributing the negative sign on the left side:

$90^{\circ} - 4x + 10^{\circ} = 40^{\circ} - x$

Next, we combine the constant terms on the left side:

$100^{\circ} - 4x = 40^{\circ} - x$

Now, we isolate the $x$ terms by adding $4x$ to both sides:

$100^{\circ} = 40^{\circ} + 3x$

Subtracting $40^{\circ}$ from both sides gives us:

$60^{\circ} = 3x$

Finally, we divide both sides by 3 to solve for $x$:

$x = 20^{\circ}$

Therefore, the value of $x$ that satisfies the given equation is $20^{\circ}$.

Considering General Solutions

While we have found a solution $x = 20^{\circ}$, it is important to consider the general solution of trigonometric equations. The cosine function has a period of $360^{\circ}$, which means that $\cos(\theta) = \cos(\theta + 360^{\circ}k)$ for any integer $k$. Therefore, if $\cos(A) = \cos(B)$, then $A = B + 360^{\circ}k$ or $A = -B + 360^{\circ}k$ for some integer $k$.

In our case, we have:

$90^{\circ} - (4x - 10)^{\circ} = (40-x)^{\circ} + 360^{\circ}k$

which we already solved to get $x = 20^{\circ}$, and

$90^{\circ} - (4x - 10)^{\circ} = -(40-x)^{\circ} + 360^{\circ}k$

Simplifying the second equation:

$100^{\circ} - 4x = -40^{\circ} + x + 360^{\circ}k$

$140^{\circ} = 5x + 360^{\circ}k$

$x = 28^{\circ} - 72^{\circ}k$

For $k = 0$, we get $x = 28^{\circ}$. However, we are looking for a specific solution among the given options. The solution $x = 20^{\circ}$ we found earlier is the one that matches the options provided.

Selecting the Correct Answer

Based on our calculations, the value of $x$ that satisfies the equation $\sin (4 x-10)^{\circ}=\cos (40-x)^{\circ}$ is $x = 20^{\circ}$. Therefore, the correct answer is:

A. $x=20$

Conclusion

Solving trigonometric equations often involves a combination of trigonometric identities and algebraic techniques. In this case, we successfully solved for $x$ by utilizing the identity $\sin(\theta) = \cos(90^{\circ} - \theta)$ and then employing algebraic manipulation to isolate $x$. Remember to always consider the general solutions when dealing with trigonometric equations, but in this instance, the specific solution $x = 20^{\circ}$ matches one of the given options.