Difference between revisions of "005 Sample Final A, Question 11"

From Grad Wiki
Jump to navigation Jump to search
(Created page with "''' Question ''' Solve the following equation in the interval <math> [0, 2\pi)</math> <br> <center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center> {|...")
 
Line 5: Line 5:
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
−
! Final Answers
+
! Step 1:
 
|-
 
|-
−
|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
+
| We need to get rid of the <math>\sin^2(\theta)</math> term. Since <math>\sin^2(\theta)=1-\cos^2(\theta)</math>, the equation becomes
 
|-
 
|-
−
|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
+
|<math>(1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) </math>.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 2:
 
|-
 
|-
−
|c) False. <math>y = x^2</math> does not have an inverse.
+
| If we simplify and move all the terms to the right hand side, we have <math>0=2\cos^2(\theta)+\cos(\theta)</math>.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 3:
 
|-
 
|-
−
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+
| Now, factoring, we have <math>0=\cos(\theta)(2\cos(\theta)+1)</math>. Thus, either <math>\cos(\theta)=0</math> or <math>2\cos(\theta)+1=0</math>.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 4:
 
|-
 
|-
−
|e) True.
+
|  
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 5:
 
|-
 
|-
−
|f) False.
+
| The solutions to <math>\cos(\theta)=0</math> in <math> [0, 2\pi)</math> are <math>\theta=\frac{\pi}{2}</math> or
 +
|-
 +
|
 +
|-
 +
|
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Final Answer:
 +
|-
 +
|
 
|}
 
|}

Revision as of 09:03, 20 May 2015

Question Solve the following equation in the interval

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)}


Step 1:
We need to get rid of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(\theta)} term. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2(\theta)=1-\cos^2(\theta)} , the equation becomes
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) } .
Step 2:
If we simplify and move all the terms to the right hand side, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0=2\cos^2(\theta)+\cos(\theta)} .
Step 3:
Now, factoring, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0=\cos(\theta)(2\cos(\theta)+1)} . Thus, either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta)=0} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\cos(\theta)+1=0} .
Step 4:
Step 5:
The solutions to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta)=0} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 2\pi)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\frac{\pi}{2}} or
Final Answer:
Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=005_Sample_Final_A,_Question_11&oldid=795"