Exercise 33.1 Prove, Without The Use Of A Calculator, That: $\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}$3.2 Determine The General Solution Of: 1 + 4 Sin ⁡ 2 X − 5 Sin ⁡ X + Cos ⁡ 2 X = 0 1 + 4 \sin^2 X - 5 \sin X + \cos 2x = 0 1 + 4 Sin 2 X − 5 Sin X + Cos 2 X = 0 (7)

Introduction

Trigonometric equations are an essential part of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will focus on solving two trigonometric equations: proving that $\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}$ and determining the general solution of $1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 0$. We will use various trigonometric identities and techniques to solve these equations.

Proving the Trigonometric Identity

3.1 Prove, without the use of a calculator, that: $\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}$

To prove this trigonometric identity, we can use the sum-to-product identity, which states that $\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$. We can apply this identity to the given equation.

import math
angle_75 = math.radians(75)
angle_15 = math.radians(15)

sum_cosines = 2 * math.cos((angle_75 + angle_15) / 2) * math.cos((angle_75 - angle_15) / 2)

simplified_expression = 2 * math.cos(math.radians(45)) * math.cos(math.radians(30))

result = simplified_expression
print(result)

Using the sum-to-product identity, we can rewrite the equation as:

$\cos 75^{\circ} + \cos 15^{\circ} = 2\cos\left(\frac{75^{\circ}+15^{\circ}}{2}\right)\cos\left(\frac{75^{\circ}-15^{\circ}}{2}\right)$

Simplifying the expression, we get:

$\cos 75^{\circ} + \cos 15^{\circ} = 2\cos 45^{\circ}\cos 30^{\circ}$

Evaluating the expression, we get:

$\cos 75^{\circ} + \cos 15^{\circ} = 2\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)$

Simplifying further, we get:

$\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}$

Therefore, we have proved that $\cos 75^{\circ} + \cos 15^{\circ} = \frac{\sqrt{6}}{2}$.

Solving the Trigonometric Equation

3.2 Determine the general solution of: $1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 0$

To solve this trigonometric equation, we can start by simplifying the equation using trigonometric identities.

import math

x = math.radians(0)

simplified_equation = 1 + 4 * math.sin(x)**2 - 5 * math.sin(x) + math.cos(2*x)

further_simplified_equation = 1 + 4 * (1 - math.cos(x)**2) - 5 * math.sin(x) + (2 * math.cos(x)**2 - 1)

even_further_simplified_equation = 4 - 4 * math.cos(x)**2 - 5 * math.sin(x) + 2 * math.cos(x)**2 - 1

final_simplified_equation = 3 - 2 * math.cos(x)**2 - 5 * math.sin(x)

factored_equation = 3 - 2 * (1 - math.sin(x)**2) - 5 * math.sin(x)

simplified_factored_equation = 3 - 2 + 2 * math.sin(x)**2 - 5 * math.sin(x)

further_simplified_factored_equation = 1 + 2 * math.sin(x)**2 - 5 * math.sin(x)

factored_equation_2 = 1 + 2 * math.sin(x)**2 - 5 * math.sin(x)

factored_equation_3 = (1 - math.sin(x)) * (2 * math.sin(x) + 1)

solution = math.pi / 6

Using the double-angle identity, we can rewrite the equation as:

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 1 + 4 \sin^2 x - 5 \sin x + 2 \cos^2 x - 1$

Simplifying the expression, we get:

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 4 \sin^2 x - 5 \sin x + 2 \cos^2 x$

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

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 4 \sin^2 x - 5 \sin x + 2 (1 - \sin^2 x)$

Simplifying the expression, we get:

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 4 \sin^2 x - 5 \sin x + 2 - 2 \sin^2 x$

Simplifying further, we get:

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = 2 \sin^2 x - 5 \sin x + 2$

Factoring out the common term, we get:

$1 + 4 \sin^2 x - 5 \sin x + \cos 2x = (1 - \sin x)(2 \sin x + 1)$

Solving the equation, we get:

$\sin x = \frac{1}{2}$

Therefore, the general solution of the equation is:

$x = \frac{\pi}{6} + 2k\pi$

where $k$ is an integer.

Conclusion

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Introduction

Trigonometric equations are an essential part of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will provide a Q&A guide to help you understand and solve trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$
• $\sec x = \frac{1}{\cos x}$
• $\csc x = \frac{1}{\sin x}$

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use various techniques, such as:

• Using trigonometric identities to simplify the equation
• Factoring out common terms
• Using the quadratic formula
• Using inverse trigonometric functions

Q: What is the difference between a trigonometric equation and a trigonometric function?

A: A trigonometric function is a function that involves trigonometric ratios, such as sine, cosine, and tangent. A trigonometric equation, on the other hand, is an equation that involves trigonometric functions.

Q: How do I determine the general solution of a trigonometric equation?

A: To determine the general solution of a trigonometric equation, you can use the following steps:

1. Simplify the equation using trigonometric identities
2. Factor out common terms
3. Use the quadratic formula
4. Use inverse trigonometric functions
5. Write the solution in terms of the trigonometric function

Q: What is the significance of the general solution of a trigonometric equation?

A: The general solution of a trigonometric equation represents all possible solutions of the equation. It is an important concept in mathematics and is used in a variety of applications, such as physics, engineering, and computer science.

Q: How do I apply trigonometric equations in real-world problems?

A: Trigonometric equations are used in a variety of real-world problems, such as:

• Calculating the height of a building or a mountain
• Determining the distance between two points
• Calculating the angle of elevation or depression
• Solving problems in physics, engineering, and computer science

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

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

• Not simplifying the equation using trigonometric identities
• Not factoring out common terms
• Not using the quadratic formula
• Not using inverse trigonometric functions
• Not writing the solution in terms of the trigonometric function

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve trigonometric equations. We hope that this article has provided a clear and concise explanation of how to solve trigonometric equations and has helped you to avoid common mistakes.