005 Sample Final A, Question 12 - Revision history

Matt Lee at 03:52, 22 May 2015

2015-05-22T03:52:50Z

Kayla Murray at 16:51, 20 May 2015

2015-05-20T16:51:54Z

Matt Lee: Created page with "''' Question ''' Given that $\sec(\theta) = -2$ and $\tan(\theta) > 0$ , find the exact values of the remaining trig functions. {| class="mw-collapsib..."

2015-05-01T04:21:03Z

Created page with "''' Question ''' Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions. {| class="mw-collapsib..."

New page

''' Question ''' Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
|-
|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>
|-
|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.
|-
|c) False. <math>y = x^2</math> does not have an inverse.
|-
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
|-
|e) True.
|-
|f) False.
|}

← Older revision		Revision as of 03:52, 22 May 2015
Line 1:		Line 1:
	''' Question ''' Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.		''' Question ''' Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.

		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	! Foundations
		+	\|-
		+	\|1) Which quadrant is <math>\theta</math> in?
		+	\|-
		+	\|2) Which trig functions are positive in this quadrant?
		+	\|-
		+	\|3) What are the side lengths of the triangle associated to <math>\theta?</math>
		+	\|-
		+	\|Answers:
		+	\|-
		+	\|1) <math>\theta</math> is in the third quadrant. We know it is in the second or third quadrant since <math>\cos</math> is negative. Since \<math>\tan</math> is positive <math>\theta</math> is in the third quadrant.
		+	\|-
		+	\|2) <math>\tan</math> and <math>\cot</math> are both positive in this quadrant. All other trig functions are negative.
		+	\|-
		+	\|3) The side lengths are 2, 1, and <math>\sqrt{3}.</math>
		+	\|}

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 3: / Line 3: @@
 {| 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>
+| Since <math>\sec(\theta)=-2</math>, we have <math>\cos(\theta)=\frac{1}{\sec(\theta)}=\frac{-1}{2}</math>.
 |-
-|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.
+| We look for solutions to <math>\theta</math> on the unit circle. The two angles on the unit circle with <math>\cos(\theta)=\frac{-1}{2}</math> are <math>\theta=\frac{2\pi}{3}</math> and <math>\theta=\frac{4\pi}{3}</math>.
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+|But, <math>\tan\left(\frac{2\pi}{3}\right)=-\sqrt{3}</math>. Since <math>\tan(\theta)>0</math>. we must have <math>\theta=\frac{4\pi}{3}</math>.
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+| The remaining values of the trig functions are
 |-
-|e) True.
+|<math>\sin(\theta)=\sin\left(\frac{4\pi}{3}\right)=\frac{-\sqrt{3}}{2}</math>,
 |-
-|f) False.
+|  <math>\tan(\theta)=\tan\left(\frac{4\pi}{3}\right)=\sqrt{3}</math>
 |}

005 Sample Final A, Question 12 - Revision history

Matt Lee at 03:52, 22 May 2015

Kayla Murray at 16:51, 20 May 2015

Matt Lee: Created page with "''' Question ''' Given that \sec(\theta) = -2 and \tan(\theta) > 0 , find the exact values of the remaining trig functions. {| class="mw-collapsib..."

Matt Lee: Created page with "''' Question ''' Given that $\sec(\theta) = -2$ and $\tan(\theta) > 0$ , find the exact values of the remaining trig functions. {| class="mw-collapsib..."