Question

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. x = 0 , y ≥ 0

Accepted Answer

Identify the geometric representation of the terminal side of the angle $$ 	heta $$ given by the equation $$ x = 0 $$ with the restriction $$ y \geq 0 . $$This corresponds to the positive y-axis in the coordinate plane. Recognize that the least positive angle $$ 	heta  in $$standard position whose terminal side lies on the positive y-axis is $$ 	heta = \frac{\pi}{2} $$ radians (or 90 degrees). Recall the definitions of the six trigonometric functions in terms of coordinates $$ (x, y)  on $$the terminal side and the radius $$ r = \sqrt{x^2 + y^2} $$:

$$ \sin 	heta = \frac{y}{r}, \quad \cos 	heta = \frac{x}{r}, \quad 	an 	heta = \frac{y}{x} $$

$$ \csc 	heta = \frac{r}{y}, \quad \sec 	heta = \frac{r}{x}, \quad \cot 	heta = \frac{x}{y} $$ Substitute the coordinates for any point on the terminal side (for example, $$ (0, y) $$ with $$ y \u003e 0 ) $$into the formulas. Since $$ x = 0 $$, calculate $$ r = \sqrt{0^2 + y^2} = y . $$Then express each trigonometric function in terms of $$ y $$ and simplify. Interpret the results carefully, noting any undefined functions due to division by zero (for example, $$ 	an 	heta = \frac{y}{0}  is $$undefined). This will give you the values of the six trigonometric functions for $$ 	heta = \frac{\pi}{2} .$$

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. x = 0 , y ≥ 0

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Key Concepts

Standard Position of an Angle

Equation of the Terminal Side and Coordinate Restrictions

Six Trigonometric Functions of an Angle