Difference between revisions of "005 Sample Final A, Question 15"

From Grad Wiki
Jump to navigation Jump to search
(Created page with "''' Question ''' Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center> {| class="mw-collapsible mw-collapsed" style = "...")
 
 
(2 intermediate revisions by one other user not shown)
Line 2: Line 2:
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
−
! Final Answers
+
! Foundations
 
|-
 
|-
−
|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>
+
|1) <math>\tan^{-1}(x)</math> can be thought of as <math>\tan^{-1}\left(\frac{x}{1}\right),</math> and this now refers to an angle in a triangle. What are the side lengths of this triangle?
 
|-
 
|-
−
|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.
+
|Answers:
 
|-
 
|-
−
|c) False. <math>y = x^2</math> does not have an inverse.
+
|1) The side lengths are 1, x, and <math>\sqrt{1 + x^2}.</math>
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 1:
 
|-
 
|-
−
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+
|First, let <math>\theta=\tan^{-1}(x)</math>. Then, <math>\tan(\theta)=x</math>.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 2:
 
|-
 
|-
−
|e) True.
+
| Now, we draw the right triangle corresponding to <math>\theta</math>. Two of the side lengths are 1 and x and the hypotenuse has length <math>\sqrt{x^2+1}</math>.
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Step 3:
 
|-
 
|-
−
|f) False.  
+
| Since <math>\cos(\theta)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}</math>, <math>\cos(\tan^{-1}(x))=\cos(\theta)=\frac{1}{\sqrt{x^2+1}}</math>.
 +
|-
 +
|
 +
|}
 +
 
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
! Final Answer:
 +
|-
 +
| <math>\frac{1}{\sqrt{x^2+1}}</math>
 
|}
 
|}

Latest revision as of 21:13, 21 May 2015

Question Find an equivalent algebraic expression for the following,

Foundations
1) can be thought of as and this now refers to an angle in a triangle. What are the side lengths of this triangle?
Answers:
1) The side lengths are 1, x, and
Step 1:
First, let . Then, .
Step 2:
Now, we draw the right triangle corresponding to . Two of the side lengths are 1 and x and the hypotenuse has length .
Step 3:
Since , .
Final Answer:
Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=005_Sample_Final_A,_Question_15&oldid=821"