Sin 2 Theta Is Equal To 0.75 What Is The Value 0f Theta

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on solving trigonometric equations, specifically the equation sin(2θ) = 0.75, to find the value of theta.

What is Sin(2θ)?

The sine function is a fundamental trigonometric function that is used to describe the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. The sine function is denoted by sin(θ) and is defined as:

sin(θ) = opposite side / hypotenuse

The double-angle formula for sine is given by:

sin(2θ) = 2sin(θ)cos(θ)

This formula allows us to express the sine of a double angle in terms of the sines and cosines of the original angle.

Solving the Equation sin(2θ) = 0.75

To solve the equation sin(2θ) = 0.75, we can use the inverse sine function, which is denoted by sin^(-1)(x). The inverse sine function returns the angle whose sine is equal to a given value.

sin^(-1)(0.75) = 2θ

To find the value of theta, we can divide both sides of the equation by 2:

θ = sin^(-1)(0.75) / 2

Using a calculator, we can find the value of sin^(-1)(0.75), which is approximately 0.8486. Therefore, we can substitute this value into the equation:

θ = 0.8486 / 2 θ = 0.4243

Finding the Value of Theta in Radians and Degrees

In mathematics, angles are often measured in radians or degrees. To find the value of theta in radians and degrees, we can use the following conversions:

1 radian = 180/π degrees 1 degree = π/180 radians

Using these conversions, we can find the value of theta in radians and degrees:

θ (radians) = 0.4243 θ (degrees) = 0.4243 × (180/π) ≈ 24.26°

Graphical Representation of the Solution

To visualize the solution, we can plot the graph of the sine function and the line y = 0.75. The point of intersection between the two curves represents the solution to the equation sin(2θ) = 0.75.

Conclusion

In this article, we have solved the trigonometric equation sin(2θ) = 0.75 to find the value of theta. We have used the inverse sine function to find the value of theta in radians and degrees. The solution to the equation is θ ≈ 0.4243 radians or θ ≈ 24.26°. This article demonstrates the importance of trigonometry in mathematics and its applications in various fields.

Code Implementation

Here is a Python code implementation to solve the equation sin(2θ) = 0.75:

import math
def solve_trig_equation():
# Define the equation
sin_2theta = 0.75
# Use the inverse sine function to find the value of 2θ
theta = math.asin(sin_2theta) / 2

# Convert the value of theta to degrees
theta_degrees = math.degrees(theta)

return theta, theta_degrees

theta, theta_degrees = solve_trig_equation()
print(f"θ (radians) = {theta}")
print(f"θ (degrees) = {theta_degrees}")

This code uses the math.asin() function to find the value of 2θ and then divides the result by 2 to find the value of theta. The math.degrees() function is used to convert the value of theta from radians to degrees.