From Förberedande kurs i matematik 1

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-	{{NAVCONTENT_START}}	+	Because the angle <math>v</math> satisfies <math>\pi \le v\le 3\pi/2\,</math>, <math>v</math> belongs to the third quadrant in the unit circle. Furthermore, <math>\tan v = 3</math> gives that the line which corresponds to the angle
-	<center> [[Bild:4_3_6c-1(2).gif]] </center>	+	<math>v</math> has slope 3.
-	{{NAVCONTENT_STOP}}	+
-	{{NAVCONTENT_START}}	+
-	<center> [[Bild:4_3_6c-2(2).gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+
-	[[Bild:4_3_6_c1.gif|center]]	+	[[Image:4_3_6_c1.gif|center]]
-	[[Bild:4_3_6_c2.gif|center]]	+
		+	In the third quadrant, we can introduce a right-angled triangle in which the hypotenuse is 1 and the sides have a 3:1 ratio.
		+
		+	[[Image:4_3_6_c2.gif|center]]
		+
		+	If we now use the Pythagorean theorem on the triangle, we see that the horizontal side ''a'' satisfies
		+
		+	{{Displayed math||<math>a^2 + (3a)^2 = 1^2</math>}}
		+
		+	which gives us that <math>10a^{2}=1</math> i.e. <math>a = 1/\!\sqrt{10}\,\textrm{.}</math>
		+
		+	Thus, the angle ''v'''s ''x''-coordinate is <math>-1/\!\sqrt{10}</math> and ''y''-coordinate is <math>-3/\!\sqrt{10}</math>, i.e.
		+
		+	{{Displayed math||<math>\begin{align}
		+	\cos v &= -\frac{1}{\sqrt{10}}\,,\\[5pt]
		+	\sin v &= -\frac{3}{\sqrt{10}}\,\textrm{.}
		+	\end{align}</math>}}

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Current revision

Because the angle \displaystyle v satisfies \displaystyle \pi \le v\le 3\pi/2\,, \displaystyle v belongs to the third quadrant in the unit circle. Furthermore, \displaystyle \tan v = 3 gives that the line which corresponds to the angle \displaystyle v has slope 3.

In the third quadrant, we can introduce a right-angled triangle in which the hypotenuse is 1 and the sides have a 3:1 ratio.

If we now use the Pythagorean theorem on the triangle, we see that the horizontal side a satisfies

\displaystyle a^2 + (3a)^2 = 1^2

which gives us that \displaystyle 10a^{2}=1 i.e. \displaystyle a = 1/\!\sqrt{10}\,\textrm{.}

Thus, the angle v's x-coordinate is \displaystyle -1/\!\sqrt{10} and y-coordinate is \displaystyle -3/\!\sqrt{10}, i.e.

\displaystyle \begin{align} \cos v &= -\frac{1}{\sqrt{10}}\,,\\[5pt] \sin v &= -\frac{3}{\sqrt{10}}\,\textrm{.} \end{align}