Question

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. tan θ = ―15/8 , and θ is in quadrant II .

Accepted Answer

Recall that the six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent, defined as follows: $$\sin 	heta = \frac{y}{r}$$, $$\cos 	heta = \frac{x}{r}$$, $$	an 	heta = \frac{y}{x}$$, $$\csc 	heta = \frac{r}{y}$$, $$\sec 	heta = \frac{r}{x}$$, and $$\cot 	heta = \frac{x}{y}$$, where $$(x, y) is a $$point on the terminal side of angle $$	heta$$ and $$r = \sqrt{x^2$ + y^2$}. $$Given $$	an 	heta = -\frac{15}{8}$$ and that $$	heta is in $$quadrant II, determine the signs of $$x$$ and $$y. In $$quadrant II, $$x is $$negative and $$y is $$positive. Since $$	an 	heta = \frac{y}{x} is $$negative, this matches the given ratio with $$y = 15$$ and $$x = -8 ($$choosing values consistent with the signs in quadrant II). Calculate the hypotenuse $$r$$ using the Pythagorean theorem: $$r = \sqrt{x^2$ + y^2$} = \sqrt{(-8)^2$ + 15^2$} = \sqrt{64 + 225}. $$Find $$\sin 	heta$$ and $$\cos 	heta$$ using the definitions: $$\sin 	heta = \frac{y}{r}$$ and $$\cos 	heta = \frac{x}{r}. $$Substitute the values of $$x$$, $$y$$, and $$r. $$Use the values of $$\sin 	heta$$ and $$\cos 	heta to $$find the remaining functions: $$	an 	heta = \frac{\sin 	heta}{\cos 	heta}$$, $$\csc 	heta = \frac{1}{\sin 	heta}$$, $$\sec 	heta = \frac{1}{\cos 	heta}$$, and $$\cot 	heta = \frac{1}{	an 	heta}. $$Rationalize denominators where necessary.

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. tan θ = ―15/8 , and θ is in quadrant II .

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Concetti chiave

Trigonometric Functions and Their Relationships

Sign of Trigonometric Functions in Quadrants

Rationalizing Denominators