Determine whether the statement is true or false. Justify...

Determine whether the statement is true or false. Justify your answer. Consider an angle in standard position with $r=12$ centimeters, as shown in the figure. Write a short paragraph describing the changes in the values of $x, y, \sin \theta, \cos \theta,$ and $\tan \theta$ as $\theta$ increases continuously from $0^{\circ}$ to $90^{\circ}$.

Determine whether the statement is true or false. Justify your answer.
Consider an angle in standard position with $r=12$ centimeters, as shown in the figure. Write a short paragraph describing the changes in the values of $x, y, \sin \theta, \cos \theta,$ and $\tan \theta$ as $\theta$ increases
continuously from $0^{\circ}$ to $90^{\circ}$.

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Key Concepts

Angle in Standard Position

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. As the angle increases, its terminal side rotates counterclockwise, and the coordinates where it intersects a circle of a given radius are determined by evaluating the cosine and sine of the angle.

Trigonometric Functions (Sine and Cosine)

Cosine and sine are defined as the x and y coordinates, respectively, of a point on a circle of fixed radius. As an angle increases from 0° to 90°, cosine (the x-coordinate) decreases from its maximum value to 0 while sine (the y-coordinate) increases from 0 to its maximum value.

Tangent Function

The tangent function is defined as the ratio of sine to cosine. As the angle increases from 0° to 90°, with sine increasing and cosine decreasing, the tangent value increases, moving from 0 toward infinity as the cosine value approaches zero.

Scaling by the Radius

When using a fixed radius, such as 12 centimeters in the example, the values of x and y are scaled by that constant. This scaling does not alter the fundamental behavior of the sine, cosine, and tangent functions; it simply multiplies the basic unit circle values by the radius.

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