After Working Through The Problem Below, Find { \cos (A)$} : : : {55^2 = 50^2 + 35^2 - 2(50)(35) \cos (A)\} { \cos (A) =$}$ { \square$}$

Introduction

In trigonometry, we often encounter equations that involve the cosine function. These equations can be used to solve for unknown angles in a right-angled triangle. In this article, we will work through a problem to find the value of cosine in a given equation.

The Problem

We are given the equation:

$$55^2 = 50^2 + 35^2 - 2(50)(35) \cos (A)$$

Our goal is to solve for $\cos (A)$.

Step 1: Expand the Squares

To start solving the equation, we need to expand the squares on both sides of the equation.

$$55^2 = 50^2 + 35^2 - 2(50)(35) \cos (A)$$

$$\Rightarrow 3025 = 2500 + 1225 - 2(50)(35) \cos (A)$$

Step 2: Simplify the Equation

Now, we can simplify the equation by combining like terms.

$$3025 = 2500 + 1225 - 2(50)(35) \cos (A)$$

$$\Rightarrow 3025 = 3725 - 3500 \cos (A)$$

Step 3: Isolate the Cosine Term

Next, we need to isolate the cosine term on one side of the equation.

$$3025 = 3725 - 3500 \cos (A)$$

$$\Rightarrow 3500 \cos (A) = 3725 - 3025$$

Step 4: Simplify the Right-Hand Side

Now, we can simplify the right-hand side of the equation.

$$3500 \cos (A) = 3725 - 3025$$

$$\Rightarrow 3500 \cos (A) = 700$$

Step 5: Solve for Cosine

Finally, we can solve for $\cos (A)$ by dividing both sides of the equation by 3500.

$$3500 \cos (A) = 700$$

$$\Rightarrow \cos (A) = \frac{700}{3500}$$

The Final Answer

Now, we can simplify the fraction to find the value of $\cos (A)$.

$$\cos (A) = \frac{700}{3500}$$

$$\Rightarrow \cos (A) = \frac{7}{35}$$

$$\Rightarrow \cos (A) = \frac{1}{5}$$

Conclusion

In this article, we worked through a problem to find the value of cosine in a given equation. We started by expanding the squares on both sides of the equation, then simplified the equation by combining like terms. We isolated the cosine term on one side of the equation, simplified the right-hand side, and finally solved for $\cos (A)$. The final answer is $\cos (A) = \frac{1}{5}$.

Applications of Trigonometry

Trigonometry has many real-world applications, including:

• Navigation: Trigonometry is used in navigation to calculate distances and directions between two points.
• Physics: Trigonometry is used in physics to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometry is used in engineering to design and build structures such as bridges, buildings, and roads.
• Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Common Trigonometric Identities

Here are some common trigonometric identities that are used to solve trigonometric equations:

• Pythagorean Identity: $\sin^2 (A) + \cos^2 (A) = 1$
• Sum and Difference Identities: $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$
• Product-to-Sum Identities: $\sin (A) \cos (B) = \frac{1}{2} (\sin (A + B) + \sin (A - B))$

Tips and Tricks

Here are some tips and tricks to help you solve trigonometric equations:

• Use the Pythagorean Identity: The Pythagorean Identity is a powerful tool for solving trigonometric equations.
• Use the Sum and Difference Identities: The Sum and Difference Identities can be used to simplify trigonometric expressions.
• Use the Product-to-Sum Identities: The Product-to-Sum Identities can be used to simplify trigonometric expressions.

Practice Problems

Here are some practice problems to help you practice solving trigonometric equations:

• Problem 1: Solve for $\cos (A)$ in the equation $64^2 = 50^2 + 30^2 - 2(50)(30) \cos (A)$.
• Problem 2: Solve for $\sin (A)$ in the equation $81^2 = 60^2 + 40^2 - 2(60)(40) \sin (A)$.
• Problem 3: Solve for $\tan (A)$ in the equation $25^2 = 20^2 + 15^2 - 2(20)(15) \tan (A)$.

Conclusion

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about trigonometry.

Q: What is Trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of triangles, particularly right triangles, and the relationships between their sides and angles.

Q: What are the Basic Trigonometric Functions?

A: The basic trigonometric functions are:

• Sine (sin): The ratio of the length of the side opposite a given angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.

Q: What is the Pythagorean Identity?

A: The Pythagorean Identity is a fundamental identity in trigonometry that states:

$$\sin^2 (A) + \cos^2 (A) = 1$$

This identity is used to solve trigonometric equations and to simplify trigonometric expressions.

Q: What are the Sum and Difference Identities?

A: The Sum and Difference Identities are a set of identities that relate the trigonometric functions of the sum and difference of two angles. They are:

• Sum Identity for Sine: $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$
• Sum Identity for Cosine: $\cos (A + B) = \cos (A) \cos (B) - \sin (A) \sin (B)$
• Difference Identity for Sine: $\sin (A - B) = \sin (A) \cos (B) - \cos (A) \sin (B)$
• Difference Identity for Cosine: $\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$

Q: What are the Product-to-Sum Identities?

A: The Product-to-Sum Identities are a set of identities that relate the trigonometric functions of the product and sum of two angles. They are:

• Product-to-Sum Identity for Sine: $\sin (A) \cos (B) = \frac{1}{2} (\sin (A + B) + \sin (A - B))$
• Product-to-Sum Identity for Cosine: $\cos (A) \cos (B) = \frac{1}{2} (\cos (A + B) + \cos (A - B))$

Q: How do I Solve Trigonometric Equations?

A: To solve trigonometric equations, you need to use the trigonometric identities and formulas to simplify the equation and isolate the variable. Here are some steps to follow:

1. Simplify the equation: Use the trigonometric identities and formulas to simplify the equation.
2. Isolate the variable: Use algebraic manipulations to isolate the variable.
3. Use trigonometric identities: Use the trigonometric identities to simplify the equation and isolate the variable.

Q: What are the Common Trigonometric Identities?

A: Here are some common trigonometric identities:

• Pythagorean Identity: $\sin^2 (A) + \cos^2 (A) = 1$
• Sum and Difference Identities: $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$
• Product-to-Sum Identities: $\sin (A) \cos (B) = \frac{1}{2} (\sin (A + B) + \sin (A - B))$

Q: How do I Use Trigonometry in Real-World Applications?

A: Trigonometry has many real-world applications, including:

• Navigation: Trigonometry is used in navigation to calculate distances and directions between two points.
• Physics: Trigonometry is used in physics to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometry is used in engineering to design and build structures such as bridges, buildings, and roads.
• Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we answered some frequently asked questions about trigonometry. We discussed the basic trigonometric functions, the Pythagorean identity, the sum and difference identities, and the product-to-sum identities. We also provided some tips and tricks for solving trigonometric equations and using trigonometry in real-world applications.