Finding Theta and Solving for X

The video tutorial demonstrates how to solve the equation 5 x sine of 3x = 2 over a given interval. It begins by suggesting a substitution of 3x with theta to simplify the problem. The tutorial then uses a graphing calculator to find the value of theta by calculating the inverse sine of 2/5. It explains that sine is positive in both the first and second quadrants, leading to two possible values for theta. The video concludes by substituting back to solve for x, providing two solutions. The process emphasizes the use of substitution and calculator functions to solve trigonometric equations efficiently.

What is the initial substitution made to simplify the equation 5 x sine of 3x = 2?

• Use a sum identity

• Use a double angle identity

• Replace sine with cosine

• Replace 3x with theta

What is the next step after setting 3x equal to theta in the equation 5 x sine theta = 2?

• Multiply both sides by 5

• Divide both sides by 5

• Add 5 to both sides

• Subtract 5 from both sides

Which tool is used to determine the value of theta when sine theta equals 2/5?

• A compass

• A protractor

• A graphing calculator

• A ruler

In which quadrants is sine theta positive?

• Third and fourth

• First and second

• Second and fourth

• First and third

How is the second possible value of theta calculated?

• By subtracting the reference angle from pi

• By doubling the reference angle

• By halving the reference angle

• By adding the reference angle to pi

What is the approximate value of the second theta in radians?

• 0.4115

• 3.1416

• 2.7301

• 1.5708

What is the final step to find the value of x after determining theta?

• Multiply by 3

• Subtract 3

• Divide by 3

• Add 3

What is the approximate value of x when 3x equals 0.4115?

• 0.2730

• 1.3650

• 0.1372

• 0.9100

What is the approximate value of x when 3x equals 2.7301?

• 1.3650

• 0.2730

• 0.9100

• 0.1372

What is the purpose of substituting back after finding theta?

• To simplify the equation

• To verify the solution

• To find the value of x

• To check the calculator