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. 2x + y = 0 , x ≥ 0

Accepted Answer

Rewrite the given equation of the terminal side in slope-intercept form to understand the line better. Starting with the equation $$2x + y = 0$$, solve for $$y to $$get $$y = -2x. $$Since the terminal side lies on the line $$y = -2x$$ with the restriction $$x \geq 0$$, consider a point on this line where $$x is $$positive. For simplicity, choose $$x = 1$$, then $$y = -2(1) = -2. $$This point $$(1, -2)$$ lies on the terminal side. Determine the angle $$	heta in $$standard position whose terminal side passes through the point $$(1, -2). $$Use the definition of tangent: $$	an(	heta) = \frac{y}{x} = \frac{-2}{1} = -2. $$Since $$x \geq 0$$ and $$y \u003c 0$$, the point is in the fourth quadrant, so $$	heta is $$the least positive angle between $$0$$ and $$2\pi$$ with $$	an(	heta) = -2. $$Calculate the six trigonometric functions of $$	heta$$ using the coordinates of the point $$(1, -2). $$First, find the radius $$r = \sqrt{x^2$ + y^2$} = \sqrt{1^2$ + (-2)^2$} = \sqrt{1 + 4} = \sqrt{5}. $$Then, use the definitions: $$\sin(	heta) = \frac{y}{r}$$, $$\cos(	heta) = \frac{x}{r}$$, $$	an(	heta) = \frac{y}{x}$$, $$\csc(	heta) = \frac{r}{y}$$, $$\sec(	heta) = \frac{r}{x}$$, and $$\cot(	heta) = \frac{x}{y}. $$Summarize the values of the six trigonometric functions based on the point and radius found, keeping in mind the signs of each function in the fourth quadrant.

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. 2x + y = 0 , x ≥ 0

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Concetti chiave

Standard Position of an Angle

Equation of a Line and Angle Determination

Six Trigonometric Functions