Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.

No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message


jsMath

Solution 4.3:3c

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 4.3:3c moved to Solution 4.3:3c: Robot: moved page)
Current revision (13:29, 9 October 2008) (edit) (undo)
m
 
(One intermediate revision not shown.)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
With the help of the Pythagorean identity, we can express <math>\cos v</math> in terms of <math>\sin v</math>,
-
<center> [[Image:4_3_3c.gif]] </center>
+
 
-
{{NAVCONTENT_STOP}}
+
{{Displayed math||<math>\cos^2 v + \sin^2 v = 1\qquad\Leftrightarrow\qquad \cos v = \pm\sqrt{1-\sin^2 v}\,\textrm{.}</math>}}
 +
 
 +
In addition, we know that the angle <math>v</math> lies between <math>-\pi/2</math>
 +
and <math>\pi/2</math>, i.e. either in the first or fourth quadrant, where angles always have a positive ''x''-coordinate (cosine value); thus, we can conclude that
 +
 
 +
{{Displayed math||<math>\cos v = \sqrt{1-\sin^2 v} = \sqrt{1-a^2}\,\textrm{.}</math>}}

Current revision

With the help of the Pythagorean identity, we can express cosv in terms of sinv,

cos2v+sin2v=1cosv=1sin2v. 

In addition, we know that the angle v lies between 2 and 2, i.e. either in the first or fourth quadrant, where angles always have a positive x-coordinate (cosine value); thus, we can conclude that

cosv=1sin2v=1a2.