Solving Trigonometric Equations And Verifying Identities A Comprehensive Guide

This article delves into various trigonometric problems, offering detailed solutions and explanations. We will explore how to solve trigonometric equations, verify trigonometric identities, and apply trigonometric principles in different scenarios. This comprehensive guide aims to enhance your understanding of trigonometry and improve your problem-solving skills.

1. Solving the Trigonometric Equation: ${\sin^2 x - 3 \cos^2 x = 0}$

In this section, we tackle the trigonometric equation ${\sin^2 x - 3 \cos^2 x = 0}$. Our primary goal is to find all possible values of x that satisfy this equation. Trigonometric equations often require a strategic approach, utilizing fundamental trigonometric identities to simplify the equation and isolate the variable. One of the most crucial identities in this context is the Pythagorean identity, which states that ${\sin^2 x + \cos^2 x = 1}$. This identity allows us to express sine in terms of cosine, or vice versa, thereby reducing the equation to a single trigonometric function. In this specific problem, we can leverage this identity to convert the equation into a form that involves only cosine, making it easier to solve.

To begin, we can rewrite the ${\sin^2 x}$ term using the Pythagorean identity. By rearranging ${\sin^2 x + \cos^2 x = 1}$, we get ${\sin^2 x = 1 - \cos^2 x}$. Substituting this into our original equation, we have:

${ 1 - \cos^2 x - 3 \cos^2 x = 0 }$

Now, we can combine the cosine terms:

${ 1 - 4 \cos^2 x = 0 }$

Next, we isolate the cosine term:

${ 4 \cos^2 x = 1 }$

${ \cos^2 x = \frac{1}{4} }$

Taking the square root of both sides, we get:

${ \cos x = \pm \frac{1}{2} }$

This gives us two separate equations to solve:

1. ${\cos x = \frac{1}{2}}$
2. ${\cos x = -\frac{1}{2}}$

For ${\cos x = \frac{1}{2}}$, we know that cosine is positive in the first and fourth quadrants. The reference angle for which cosine equals ${\frac{1}{2}}$ is ${\frac{\pi}{3}}$ or 60°. Therefore, the solutions in the interval ${[0, 2\pi)}$ are:

${ x = \frac{\pi}{3}, \quad x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} }$

Converting these to degrees, we have 60° and 300°.

For ${\cos x = -\frac{1}{2}}$, cosine is negative in the second and third quadrants. The reference angle remains ${\frac{\pi}{3}}$ or 60°. The solutions in the interval ${[0, 2\pi)}$ are:

${ x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}, \quad x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} }$

Converting these to degrees, we have 120° and 240°.

Therefore, the solutions to the equation ${\sin^2 x - 3 \cos^2 x = 0}$ in the interval ${[0, 2\pi)}$ are 60°, 120°, 240°, and 300°. These solutions represent the angles at which the given trigonometric equation holds true. The strategic use of the Pythagorean identity and careful consideration of the quadrants where cosine is positive or negative are crucial steps in arriving at the correct solutions.

2. Verifying the Trigonometric Identity: ${\sin(50^\circ) = \cos(90^\circ - 40^\circ)}$

In this section, we aim to verify the trigonometric identity ${\sin(50^\circ) = \cos(90^\circ - 40^\circ)}$. Verifying trigonometric identities is a fundamental skill in trigonometry, often requiring the application of various trigonometric identities and relationships. The co-function identity is particularly relevant here, as it directly relates sine and cosine functions of complementary angles. Understanding and applying these identities is crucial for simplifying and proving trigonometric expressions.

The co-function identities state that the sine of an angle is equal to the cosine of its complement, and vice versa. Specifically, we have:

${ \sin(\theta) = \cos(90^\circ - \theta) }$

${ \cos(\theta) = \sin(90^\circ - \theta) }$

These identities are derived from the geometric relationships in a right-angled triangle, where the two acute angles are complementary (add up to 90°). The sine of one acute angle is equal to the ratio of the opposite side to the hypotenuse, while the cosine of the other acute angle is equal to the ratio of the adjacent side to the hypotenuse. Since the opposite side of one angle is the adjacent side of the other, their ratios are equal.

To verify the given identity, we can apply the co-function identity directly. We have:

${ \sin(50^\circ) = \cos(90^\circ - 50^\circ) }$

Simplifying the expression inside the cosine function, we get:

${ \cos(90^\circ - 50^\circ) = \cos(40^\circ) }$

However, the original identity given is ${\sin(50^\circ) = \cos(90^\circ - 40^\circ)}$, which seems to have a slight error. Let's correct the expression inside the cosine function:

${ \cos(90^\circ - 40^\circ) = \cos(50^\circ) }$

This does not directly verify the identity ${\sin(50^\circ) = \cos(90^\circ - 40^\circ)}$. Instead, it shows that ${\cos(90^\circ - 40^\circ) = \cos(50^\circ)}$. To correctly verify the original identity, we should use the co-function identity ${\sin(\theta) = \cos(90^\circ - \theta)}$. Applying this to our problem:

${ \sin(50^\circ) = \cos(90^\circ - 50^\circ) }$

${ \sin(50^\circ) = \cos(40^\circ) }$

Now, let's consider the given expression ${\cos(90^\circ - 40^\circ)}$:

${ \cos(90^\circ - 40^\circ) = \cos(50^\circ) }$

Thus, the identity we need to verify is:

${ \sin(50^\circ) = \cos(50^\circ) }$

This identity is not true in general. ${\sin(50^\circ)}$ and ${\cos(50^\circ)}$ have different values. However, if the original problem was intended to be ${\sin(50^\circ) = \cos(40^\circ)}$, then it is indeed true, as we have shown using the co-function identity.

In conclusion, the original identity ${\sin(50^\circ) = \cos(90^\circ - 40^\circ)}$ simplifies to ${\sin(50^\circ) = \cos(50^\circ)}$, which is false. The correct identity, based on the co-function relationship, is ${\sin(50^\circ) = \cos(40^\circ)}$.

3. Simplifying the Trigonometric Expression: ${\frac{1 - \cos^2 x}{4 \cos(90^\circ + 2x)} + 0.01}$

In this section, we focus on simplifying the trigonometric expression ${\frac{1 - \cos^2 x}{4 \cos(90^\circ + 2x)} + 0.01}$. Simplifying trigonometric expressions often involves applying trigonometric identities to reduce the complexity of the expression. The Pythagorean identity and the angle sum/difference identities are particularly useful in these scenarios. Our goal is to rewrite the given expression in its simplest form by strategically using these identities.

To begin, let's consider the numerator of the fraction, ${1 - \cos^2 x}$. From the Pythagorean identity, we know that ${\sin^2 x + \cos^2 x = 1}$. Rearranging this identity, we get:

${ \sin^2 x = 1 - \cos^2 x }$

Thus, we can replace ${1 - \cos^2 x}$ in our expression with ${\sin^2 x}$.

Now, let's look at the denominator, ${4 \cos(90^\circ + 2x)}$. We can use the cosine angle sum identity, which states:

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

Applying this identity to ${\cos(90^\circ + 2x)}$, where A = 90° and B = 2x, we get:

${ \cos(90^\circ + 2x) = \cos(90^\circ) \cos(2x) - \sin(90^\circ) \sin(2x) }$

We know that ${\cos(90^\circ) = 0}$ and ${\sin(90^\circ) = 1}$. Substituting these values, we have:

${ \cos(90^\circ + 2x) = 0 \cdot \cos(2x) - 1 \cdot \sin(2x) }$

${ \cos(90^\circ + 2x) = -\sin(2x) }$

Now, we can substitute this back into our original expression:

${ \frac{1 - \cos^2 x}{4 \cos(90^\circ + 2x)} + 0.01 = \frac{\sin^2 x}{4(-\sin(2x))} + 0.01 }$

We also know the double angle identity for sine:

${ \sin(2x) = 2 \sin x \cos x }$

Substituting this into our expression, we get:

${ \frac{\sin^2 x}{4(-\sin(2x))} + 0.01 = \frac{\sin^2 x}{4(-2 \sin x \cos x)} + 0.01 }$

Simplifying further:

${ \frac{\sin^2 x}{-8 \sin x \cos x} + 0.01 = -\frac{\sin x}{8 \cos x} + 0.01 }$

We can rewrite ${\frac{\sin x}{\cos x}}$ as ${\tan x}$:

${ -\frac{\sin x}{8 \cos x} + 0.01 = -\frac{1}{8} \tan x + 0.01 }$

Thus, the simplified form of the given trigonometric expression is:

${ -\frac{1}{8} \tan x + 0.01 }$

This simplified expression is easier to work with and provides a clearer understanding of the relationship between x and the value of the expression. By applying trigonometric identities systematically, we were able to reduce a complex expression to a more manageable form.

4. Analyzing the Equation ${4 \sin^2 x - 3 = 0}$ for ${0^\circ \leq x \leq 360^\circ}$

In this section, we explore the equation ${4 \sin^2 x - 3 = 0}$ within the interval ${0^\circ \leq x \leq 360^\circ}$. Our objectives are to determine the number of quadrants in which solutions exist and to find these solutions. This involves algebraic manipulation of the equation and an understanding of the sine function's behavior in different quadrants.

First, let's solve the equation for ${\sin x}$:

${ 4 \sin^2 x - 3 = 0 }$

${ 4 \sin^2 x = 3 }$

${ \sin^2 x = \frac{3}{4} }$

Taking the square root of both sides, we get:

${ \sin x = \pm \frac{\sqrt{3}}{2} }$

This gives us two equations to consider:

1. ${\sin x = \frac{\sqrt{3}}{2}}$
2. ${\sin x = -\frac{\sqrt{3}}{2}}$

4.1. Determining the Number of Quadrants with Solutions

The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. Therefore:

• ${\sin x = \frac{\sqrt{3}}{2}}$ has solutions in the first and second quadrants.
• ${\sin x = -\frac{\sqrt{3}}{2}}$ has solutions in the third and fourth quadrants.

Thus, there will be solutions in all four quadrants.

4.2. Finding the Solutions in the Given Interval

For ${\sin x = \frac{\sqrt{3}}{2}}$, the reference angle is the angle whose sine is ${\frac{\sqrt{3}}{2}}$. This angle is 60° or ${\frac{\pi}{3}}$ radians.

• In the first quadrant, the solution is simply the reference angle: ${x = 60^\circ}$.
• In the second quadrant, the solution is ${180^\circ - 60^\circ = 120^\circ}$.

For ${\sin x = -\frac{\sqrt{3}}{2}}$, the reference angle is still 60°.

• In the third quadrant, the solution is ${180^\circ + 60^\circ = 240^\circ}$.
• In the fourth quadrant, the solution is ${360^\circ - 60^\circ = 300^\circ}$.

Therefore, the solutions to the equation ${4 \sin^2 x - 3 = 0}$ in the interval ${0^\circ \leq x \leq 360^\circ}$ are 60°, 120°, 240°, and 300°. These solutions cover all four quadrants, demonstrating the complete set of angles that satisfy the given equation within the specified interval.

In summary, this comprehensive exploration of trigonometric equations and identities provides a solid foundation for solving various trigonometric problems. By understanding and applying fundamental trigonometric identities and principles, one can effectively tackle complex equations and simplify expressions. This article serves as a valuable resource for anyone looking to enhance their knowledge and skills in trigonometry.