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 10:03, 20 May 2015

Question Solve the following equation in the interval


Step 1:
We need to get rid of the term. Since , the equation becomes
.
Step 2:
If we simplify and move all the terms to the right hand side, we have .
Step 3:
Now, factoring, we have . Thus, either or .
Step 4:
Step 5:
The solutions to in are or
Final Answer:
Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=005_Sample_Final_A,_Question_11&oldid=795"