Solving Trigonometric Equations

The video tutorial explains how to find the two smallest positive radian solutions to the trigonometric equation 4 sin(2θ) = 3. Instead of substituting for sin(2θ), the instructor substitutes 2θ with β, leading to the equation 4 sin(β) = 3. The solution involves using a calculator to find β, considering the positive sine value in the first and second quadrants, and then calculating θ by dividing β by 2. The final solutions for θ are approximately 0.424 and 1.147 radians.

What substitution is made to simplify the equation 4 sin(2θ) = 3?

• δ = 4θ

• γ = 3θ

• α = θ

• β = 2θ

What is the first step in solving the equation 4 sin(β) = 3?

• Add 3 to both sides

• Divide both sides by 4

• Subtract 3 from both sides

• Multiply both sides by 4

What is the purpose of using a calculator in this problem?

• To find the sine of 3/4

• To find the inverse sine of 3/4

• To find the tangent of β

• To find the cosine of β

Why must the solutions for β be in the first or second quadrant?

• Because tangent is negative in these quadrants

• Because sine is positive in these quadrants

• Because sine is negative in these quadrants

• Because cosine is positive in these quadrants

What is the approximate value of β1 found using the calculator?

• 2.2935

• 1.147

• 0.424

• 0.8481

How is the second solution for β in the second quadrant calculated?

• By dividing 0.8481 by 2

• By adding 0.8481 to π

• By subtracting 0.8481 from π

• By multiplying 0.8481 by 2

What is the value of θ1 after dividing β1 by 2?

• 0.424

• 1.147

• 0.8481

• 2.2935

What is the value of θ2 after dividing β2 by 2?

• 0.8481

• 1.147

• 0.424

• 2.2935

What are the two smallest positive radian solutions for θ?

• 0.424 and 1.147

• 0.8481 and 2.2935

• 1.147 and 2.2935

• 0.424 and 0.8481

What is the final step to find the solutions for θ?

• Subtract π from β

• Multiply β by 2

• Add π to β

• Divide β by 2