Solve each problem. In the following equation, t is time (in...

Solve each problem. In the following equation, $t$ is time (in seconds) and $y$ is the angle formed by a rhythmically moving arm. $$ y=\frac{1}{3} \sin \frac{4 \pi t}{3} $$ (a) Solve the equation for $t$ (b) At what time, to the nearest hundredth of a second, does the arm first form an angle of 0.3 radian?

Solve each problem.
In the following equation, $t$ is time (in seconds) and $y$ is the angle formed by a rhythmically moving arm.
$$
y=\frac{1}{3} \sin \frac{4 \pi t}{3}
$$
(a) Solve the equation for $t$
(b) At what time, to the nearest hundredth of a second, does the arm first form an angle of 0.3 radian?

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Key Concepts

Solving Trigonometric Equations

This concept involves isolating the trigonometric function and applying the appropriate inverse function to obtain specific solutions. Once the basic solution is found, the general solution is obtained by incorporating the periodic properties of the trigonometric function.

Periodicity of Trigonometric Functions

Understanding periodicity is key when solving equations involving sine or cosine. The sine function repeats every 2?, but a coefficient in the argument scales the period. This adjusted period must be used in the general solution to account for all possible solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin for sine equations, are used to determine the principal value or the initial solution for the angle. This initial solution serves as a starting point for deriving the full set of solutions based on the function's periodicity.

Numerical Approximation and Rounding

In many practical problems, the exact analytical solution needs to be converted to a numerical approximation. This involves using a calculator to evaluate expressions and then rounding to a specified degree of accuracy, ensuring the answer meets the problem's requirements.

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