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. ―5x ― 3y = 0 , x ≤ 0

Accepted Answer

Rewrite the given equation of the terminal side in slope-intercept form to understand the line better. Starting with $$-5x - 3y = 0$$, solve for $$y to $$get $$y = -\frac{5}{3}x. $$Since the terminal side is a ray starting from the origin (standard position), and the restriction is $$x \leq 0$$, focus on the portion of the line where $$x is $$non-positive (to the left of the origin). Determine the angle $$	heta$$ that this terminal side makes with the positive x-axis. Recall that the slope $$m of $$the line is related to the angle by $$m = 	an(	heta). $$Here, $$m = -\frac{5}{3}$$, so $$	heta = \arctan\left(-\frac{5}{3}ight). $$Since $$x \leq 0$$, the terminal side lies in either the second or third quadrant. Adjust the angle $$	heta$$ accordingly to find the least positive angle in standard position that corresponds to this ray. Once $$	heta is $$identified, use the definitions of the six trigonometric functions in terms of $$x$$ and $$y$$ coordinates on the terminal side: $$\sin 	heta = \frac{y}{r}$$, $$\cos 	heta = \frac{x}{r}$$, $$	an 	heta = \frac{y}{x}$$, $$\csc 	heta = \frac{r}{y}$$, $$\sec 	heta = \frac{r}{x}$$, and $$\cot 	heta = \frac{x}{y}$$, where $$r = \sqrt{x^2$ + y^2$}. $$Substitute values consistent with the line and restriction to find these ratios.

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. ―5x ― 3y = 0 , x ≤ 0

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Key Concepts

Standard Position of an Angle

Equation of a Line and Its Relation to the Terminal Side

Six Trigonometric Functions and Their Calculation