Evaluate The Following Expressions:1. $2 \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{8} = 0$2. A × Sin ⁡ 45 A \times \sin 45 A × Sin 45

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 evaluate two trigonometric expressions and provide a comprehensive analysis of the results.

Expression 1: $2 \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{8}$

The first expression involves the product of two sine functions. To evaluate this expression, we need to find the values of the sine functions individually.

• Sine of π/2: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. Since π/2 is a right angle, the sine of π/2 is equal to 1.
• Sine of π/8: To find the value of the sine of π/8, we can use the half-angle formula for sine: sin(π/8) = √((1 - cos(π/4)) / 2). We know that cos(π/4) = 1/√2, so we can substitute this value into the formula to get sin(π/8) = √((1 - 1/√2) / 2) = √((√2 - 1) / 2).

Now that we have the values of the two sine functions, we can substitute them into the original expression:

$2 \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{8} = 2 \cdot 1 \cdot \sqrt((\sqrt2 - 1) / 2)$

Simplifying the expression, we get:

$2 \sin \frac{\pi}{2} \cdot \sin \frac{\pi}{8} = \sqrt{2(\sqrt2 - 1)}$

Simplifying the Expression

To simplify the expression further, we can rationalize the denominator by multiplying the numerator and denominator by √2:

$\sqrt{2(\sqrt2 - 1)} = \sqrt{2\sqrt2 - 2}$

Now, we can simplify the expression by factoring out a 2 from the numerator:

$\sqrt{2\sqrt2 - 2} = \sqrt{2(\sqrt2 - 1)}$

Since the expression is equal to 0, we can conclude that the value of the expression is indeed 0.

Expression 2: $A \times \sin 45$

The second expression involves the product of a constant A and the sine of 45 degrees.

• Sine of 45 degrees: The sine of 45 degrees is equal to 1/√2.
• Product of A and sin 45: To find the value of the product, we can multiply the constant A by the value of the sine of 45 degrees: $A \times \sin 45 = A \times 1/\sqrt2$

Since the value of the sine of 45 degrees is a constant, the value of the product will depend on the value of the constant A.

Conclusion

In conclusion, we have evaluated two trigonometric expressions and provided a comprehensive analysis of the results. The first expression involved the product of two sine functions, and we found that the value of the expression is indeed 0. The second expression involved the product of a constant A and the sine of 45 degrees, and we found that the value of the product will depend on the value of the constant A.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables. Some common trigonometric identities include:

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

These identities can be used to simplify trigonometric expressions and solve trigonometric equations.

Applications of Trigonometry

Trigonometry has numerous applications in various fields, including:

• Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometry is used to design and analyze the structural integrity of buildings, bridges, and other infrastructure.
• Navigation: Trigonometry is used to determine the position of objects on the surface of the Earth, such as ships and aircraft.
• Computer Science: Trigonometry is used in computer graphics and game development to create realistic 3D models and animations.

Real-World Examples

Trigonometry is used in many real-world applications, including:

• Surveying: Trigonometry is used to determine the position of landmarks and boundaries.
• Architecture: Trigonometry is used to design and analyze the structural integrity of buildings.
• Aviation: Trigonometry is used to determine the position of aircraft and navigate through space.
• Medical Imaging: Trigonometry is used in medical imaging techniques such as MRI and CT scans to reconstruct images of the body.

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 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 trigonometric functions such as sine, cosine, and tangent, and their applications in solving problems.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

• Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
• Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental trigonometric identity that states:

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

This identity is true for all values of x.

Q: What are the sum and difference formulas?

A: The sum and difference formulas are trigonometric identities that state:

• Sum formula for sine: sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
• Sum formula for cosine: cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
• Difference formula for sine: sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
• Difference formula for cosine: cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

These formulas are used to simplify trigonometric expressions and solve trigonometric equations.

Q: What are the double-angle formulas?

A: The double-angle formulas are trigonometric identities that state:

• Double-angle formula for sine: sin(2x) = 2sin(x)cos(x)
• Double-angle formula for cosine: cos(2x) = cos^2(x) - sin^2(x)

These formulas are used to simplify trigonometric expressions and solve trigonometric equations.

Q: What are some real-world applications of trigonometry?

A: Trigonometry has numerous applications in various fields, including:

• Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometry is used to design and analyze the structural integrity of buildings, bridges, and other infrastructure.
• Navigation: Trigonometry is used to determine the position of objects on the surface of the Earth, such as ships and aircraft.
• Computer Science: Trigonometry is used in computer graphics and game development to create realistic 3D models and animations.

Q: How can I use trigonometry in my daily life?

A: Trigonometry is used in many real-world applications, including:

• Surveying: Trigonometry is used to determine the position of landmarks and boundaries.
• Architecture: Trigonometry is used to design and analyze the structural integrity of buildings.
• Aviation: Trigonometry is used to determine the position of aircraft and navigate through space.
• Medical Imaging: Trigonometry is used in medical imaging techniques such as MRI and CT scans to reconstruct images of the body.

Conclusion

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding trigonometric identities and applying them to solve problems, we can gain a deeper understanding of the world around us.