$\sin \theta = \frac{7 \sin 60}{8}$

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Introduction

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Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific type of trigonometric equation, namely the equation $\sin \theta = \frac{7 \sin 60}{8}$. We will break down the solution into manageable steps, using various trigonometric identities and properties to simplify the equation and find the value of $\theta$.

Understanding the Equation

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The given equation is $\sin \theta = \frac{7 \sin 60}{8}$. To solve this equation, we need to isolate the variable $\theta$ and find its value. The first step is to simplify the right-hand side of the equation by evaluating the value of $\sin 60$.

Evaluating $\sin 60$

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The value of $\sin 60$ is a well-known trigonometric value, which is equal to $\frac{\sqrt{3}}{2}$. Substituting this value into the equation, we get:

$$\sin \theta = \frac{7 \cdot \frac{\sqrt{3}}{2}}{8}$$

Simplifying the Equation

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To simplify the equation further, we can multiply both the numerator and the denominator by 2 to eliminate the fraction:

$$\sin \theta = \frac{7 \sqrt{3}}{16}$$

Using Trigonometric Identities

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The next step is to use trigonometric identities to simplify the equation. One of the most useful identities in trigonometry is the Pythagorean identity, which states that $\sin^2 \theta + \cos^2 \theta = 1$. We can use this identity to rewrite the equation in terms of $\cos \theta$.

Rewriting the Equation

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Using the Pythagorean identity, we can rewrite the equation as:

$$\sin \theta = \frac{7 \sqrt{3}}{16} \Rightarrow \cos^2 \theta = 1 - \sin^2 \theta$$

Simplifying the Equation Further

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Substituting the value of $\sin \theta$ into the equation, we get:

$$\cos^2 \theta = 1 - \left(\frac{7 \sqrt{3}}{16}\right)^2$$

Evaluating the Expression

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To evaluate the expression, we need to simplify the square of the fraction:

$$\left(\frac{7 \sqrt{3}}{16}\right)^2 = \frac{49 \cdot 3}{256} = \frac{147}{256}$$

Substituting the Value

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Substituting the value into the equation, we get:

$$\cos^2 \theta = 1 - \frac{147}{256}$$

Simplifying the Equation

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To simplify the equation further, we can multiply both the numerator and the denominator by 256 to eliminate the fraction:

$$\cos^2 \theta = \frac{256 - 147}{256}$$

Evaluating the Expression

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To evaluate the expression, we need to subtract 147 from 256:

$$256 - 147 = 109$$

Substituting the Value

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Substituting the value into the equation, we get:

$$\cos^2 \theta = \frac{109}{256}$$

Finding the Value of $\theta$

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To find the value of $\theta$, we need to take the square root of both sides of the equation:

$$\cos \theta = \pm \sqrt{\frac{109}{256}}$$

Evaluating the Expression

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To evaluate the expression, we need to simplify the square root:

$$\sqrt{\frac{109}{256}} = \frac{\sqrt{109}}{\sqrt{256}} = \frac{\sqrt{109}}{16}$$

Substituting the Value

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Substituting the value into the equation, we get:

$$\cos \theta = \pm \frac{\sqrt{109}}{16}$$

Finding the Value of $\theta$

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To find the value of $\theta$, we need to use the inverse cosine function:

$$\theta = \cos^{-1} \left(\pm \frac{\sqrt{109}}{16}\right)$$

Evaluating the Expression

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To evaluate the expression, we need to use a calculator or a trigonometric table to find the value of $\theta$.

Conclusion

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In this article, we solved the trigonometric equation $\sin \theta = \frac{7 \sin 60}{8}$ using various trigonometric identities and properties. We simplified the equation step by step, using the Pythagorean identity to rewrite the equation in terms of $\cos \theta$. Finally, we used the inverse cosine function to find the value of $\theta$. The solution to the equation is $\theta = \cos^{-1} \left(\pm \frac{\sqrt{109}}{16}\right)$.

Final Answer

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The final answer is $\boxed{\cos^{-1} \left(\pm \frac{\sqrt{109}}{16}\right)}$.

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Q&A: Solving Trigonometric Equations

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Q: What is a trigonometric equation?

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A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the properties of waves.

Q: How do I solve a trigonometric equation?

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A: To solve a trigonometric equation, you need to isolate the variable (usually x or θ) and find its value. This can be done using various trigonometric identities and properties, such as the Pythagorean identity, the sum and difference formulas, and the double-angle formulas.

Q: What is the Pythagorean identity?

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A: The Pythagorean identity is a fundamental trigonometric identity that states: sin^2(x) + cos^2(x) = 1. This identity can be used to rewrite trigonometric equations in terms of sine and cosine.

Q: How do I use the Pythagorean identity to solve a trigonometric equation?

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A: To use the Pythagorean identity to solve a trigonometric equation, you need to rewrite the equation in terms of sine and cosine. Then, you can use the Pythagorean identity to simplify the equation and isolate the variable.

Q: What is the inverse cosine function?

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A: The inverse cosine function is a function that takes the cosine of an angle as input and returns the angle itself as output. This function is denoted by cos^-1(x) and is used to find the value of an angle given its cosine.

Q: How do I use the inverse cosine function to solve a trigonometric equation?

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A: To use the inverse cosine function to solve a trigonometric equation, you need to rewrite the equation in terms of cosine. Then, you can use the inverse cosine function to find the value of the angle.

Q: What are some common trigonometric identities?

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A: Some common trigonometric identities include:

• The Pythagorean identity: sin^2(x) + cos^2(x) = 1
• The sum and difference formulas: sin(x + y) = sin(x)cos(y) + cos(x)sin(y), cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
• The double-angle formulas: sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x)

Q: How do I choose the correct trigonometric identity to use in a problem?

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A: To choose the correct trigonometric identity to use in a problem, you need to analyze the equation and identify the trigonometric functions involved. Then, you can select the identity that best matches the equation and use it to simplify and solve the equation.

Q: What are some common mistakes to avoid when solving trigonometric equations?

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A: Some common mistakes to avoid when solving trigonometric equations include:

• Not using the correct trigonometric identity
• Not simplifying the equation enough
• Not isolating the variable correctly
• Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

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A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What are some real-world applications of trigonometric equations?

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A: Some real-world applications of trigonometric equations include:

• Modeling the motion of objects, such as the trajectory of a projectile or the motion of a pendulum
• Analyzing the behavior of electrical circuits, such as the voltage and current in a circuit
• Studying the properties of waves, such as the amplitude and frequency of a wave
• Solving problems in navigation, such as finding the distance and direction between two points on the Earth's surface.