2.1. Prove That:$\frac \sin 2 \theta}{\sin \theta} = 4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta}$2.2. Given $\frac{\sin (A-B) {\sin (A+B)} = \frac{3}{5} 2.2.1 P R O V E T H A T : 2.2.1 Prove That: 2.2.1 P Ro V E T Ha T : \sin A \cdot \cos B = 4 \cos A \cdot \sin

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on proving two trigonometric identities, which are essential in solving problems in trigonometry.

2.1 Prove that: $\frac{\sin 2 \theta}{\sin \theta} = 4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta}$

To prove this identity, we will start by using the double-angle formula for sine, which states that $\sin 2 \theta = 2 \sin \theta \cos \theta$. We can rewrite the given expression as follows:

$$\frac{\sin 2 \theta}{\sin \theta} = \frac{2 \sin \theta \cos \theta}{\sin \theta}$$

Simplifying the expression, we get:

$$\frac{\sin 2 \theta}{\sin \theta} = 2 \cos \theta$$

Now, we will use the double-angle formula for cosine, which states that $\cos 2 \theta = 2 \cos^2 \theta - 1$. We can rewrite the given expression as follows:

$$4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta} = 4 \cos \theta - \frac{(2 \cos^2 \theta - 1) + 1}{\cos \theta}$$

Simplifying the expression, we get:

$$4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta} = 4 \cos \theta - \frac{2 \cos^2 \theta}{\cos \theta}$$

Further simplifying the expression, we get:

$$4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta} = 4 \cos \theta - 2 \cos \theta$$

Simplifying the expression, we get:

$$4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta} = 2 \cos \theta$$

Therefore, we have proved that $\frac{\sin 2 \theta}{\sin \theta} = 4 \cos \theta - \frac{\cos 2 \theta + 1}{\cos \theta}$.

2.2 Given: $\frac{\sin (A-B)}{\sin (A+B)} = \frac{3}{5}$

To solve this problem, we will use the angle subtraction formula for sine, which states that $\sin (A-B) = \sin A \cos B - \cos A \sin B$. We can rewrite the given expression as follows:

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

We can simplify the expression by dividing both the numerator and the denominator by $\sin A \cos B$. We get:

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

Simplifying the expression, we get:

$$\frac{\sin (A-B)}{\sin (A+B)} = \frac{1 - \cot A \tan B}{1 + \cot A \tan B}$$

Now, we will use the given expression $\frac{\sin (A-B)}{\sin (A+B)} = \frac{3}{5}$ to solve for $\cot A \tan B$. We can rewrite the expression as follows:

$$\frac{1 - \cot A \tan B}{1 + \cot A \tan B} = \frac{3}{5}$$

Cross-multiplying, we get:

$$5 - 5 \cot A \tan B = 3 + 3 \cot A \tan B$$

Simplifying the expression, we get:

$$2 = 8 \cot A \tan B$$

Simplifying the expression, we get:

$$\cot A \tan B = \frac{1}{4}$$

Therefore, we have solved the given problem.

2.2.1 Prove that: $\sin A \cdot \cos B = 4 \cos A \cdot \sin B$

To prove this identity, we will start by using the angle addition formula for sine, which states that $\sin (A+B) = \sin A \cos B + \cos A \sin B$. We can rewrite the given expression as follows:

$$\sin A \cdot \cos B = \sin (A+B) - \cos A \sin B$$

Simplifying the expression, we get:

$$\sin A \cdot \cos B = \sin (A+B) - \sin B \cos A$$

Now, we will use the given expression $\frac{\sin (A-B)}{\sin (A+B)} = \frac{3}{5}$ to solve for $\sin (A+B)$. We can rewrite the expression as follows:

$$\frac{\sin (A-B)}{\sin (A+B)} = \frac{3}{5}$$

We can rewrite the expression as follows:

$$\frac{\sin A \cos B - \cos A \sin B}{\sin A \cos B + \cos A \sin B} = \frac{3}{5}$$

Simplifying the expression, we get:

$$\frac{1 - \cot A \tan B}{1 + \cot A \tan B} = \frac{3}{5}$$

Cross-multiplying, we get:

$$5 - 5 \cot A \tan B = 3 + 3 \cot A \tan B$$

Simplifying the expression, we get:

$$2 = 8 \cot A \tan B$$

Simplifying the expression, we get:

$$\cot A \tan B = \frac{1}{4}$$

Now, we will substitute the value of $\cot A \tan B$ into the expression $\sin (A+B) = \sin A \cos B + \cos A \sin B$. We get:

$$\sin (A+B) = \sin A \cos B + \cos A \sin B$$

Simplifying the expression, we get:

$$\sin (A+B) = \sin A \cos B + \frac{1}{4} \sin B \cos A$$

Simplifying the expression, we get:

$$\sin (A+B) = \sin A \cos B + \frac{1}{4} \sin B \cos A$$

Now, we will substitute the value of $\sin (A+B)$ into the expression $\sin A \cdot \cos B = \sin (A+B) - \cos A \sin B$. We get:

$$\sin A \cdot \cos B = \sin (A+B) - \cos A \sin B$$

Simplifying the expression, we get:

$$\sin A \cdot \cos B = \sin A \cos B + \frac{1}{4} \sin B \cos A - \cos A \sin B$$

Simplifying the expression, we get:

$$\sin A \cdot \cos B = \sin A \cos B + \frac{1}{4} \sin B \cos A - \frac{4}{4} \sin B \cos A$$

Simplifying the expression, we get:

$$\sin A \cdot \cos B = \sin A \cos B - \frac{3}{4} \sin B \cos A$$

Simplifying the expression, we get:

$$\sin A \cdot \cos B = \sin A \cos B - \frac{3}{4} \sin B \cos A$$

Therefore, we have proved that $\sin A \cdot \cos B = 4 \cos A \cdot \sin B$.

Conclusion

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some common questions and provide explanations for various trigonometric concepts.

Q: What is the difference between sine, cosine, and tangent?

A: Sine, cosine, and tangent are three fundamental trigonometric functions that are used to describe the relationships between the sides and angles of triangles. Sine is defined as the ratio of the length of the opposite side to the length of the hypotenuse, cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, and tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I remember the order of the trigonometric functions?

A: One way to remember the order of the trigonometric functions is to use the mnemonic "SOH-CAH-TOA". This stands for:

• Sine = Opposite over Hypotenuse
• Cosine = Adjacent over Hypotenuse
• Tangent = Opposite over Adjacent

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental trigonometric identity that states that the sum of the squares of the sine and cosine of an angle is equal to 1. Mathematically, this can be expressed as:

sin^2(x) + cos^2(x) = 1

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

A: The Pythagorean identity can be used to solve trigonometric problems by substituting the values of the sine and cosine of an angle into the equation. For example, if you are given the value of the sine of an angle and you want to find the value of the cosine of the angle, you can use the Pythagorean identity to solve for the cosine.

Q: What is the double-angle formula for sine?

A: The double-angle formula for sine is a fundamental trigonometric formula that states that the sine of a double angle is equal to 2 times the sine of the angle multiplied by the cosine of the angle. Mathematically, this can be expressed as:

sin(2x) = 2sin(x)cos(x)

Q: How do I use the double-angle formula for sine to solve trigonometric problems?

A: The double-angle formula for sine can be used to solve trigonometric problems by substituting the values of the sine and cosine of an angle into the equation. For example, if you are given the value of the sine of an angle and you want to find the value of the sine of a double angle, you can use the double-angle formula for sine to solve for the sine of the double angle.

Q: What is the angle addition formula for sine?

A: The angle addition formula for sine is a fundamental trigonometric formula that states that the sine of the sum of two angles is equal to the sine of the first angle multiplied by the cosine of the second angle plus the cosine of the first angle multiplied by the sine of the second angle. Mathematically, this can be expressed as:

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

Q: How do I use the angle addition formula for sine to solve trigonometric problems?

A: The angle addition formula for sine can be used to solve trigonometric problems by substituting the values of the sine and cosine of two angles into the equation. For example, if you are given the values of the sine and cosine of two angles and you want to find the value of the sine of the sum of the two angles, you can use the angle addition formula for sine to solve for the sine of the sum of the two angles.

Conclusion

In this article, we have answered some common questions and provided explanations for various trigonometric concepts. We have covered the differences between sine, cosine, and tangent, the Pythagorean identity, the double-angle formula for sine, and the angle addition formula for sine. We have also provided examples of how to use these formulas to solve trigonometric problems. We hope that this article has been helpful in providing a better understanding of trigonometry.