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

Latest revision as of 20:43, 21 May 2015

Question Solve the following equation in the interval $[0,2\pi )$

\sin ^{2}(\theta )-\cos ^{2}(\theta )=1+\cos(\theta )

Foundations:
1) Which trigonometric identities are useful in this problem?
Answer:
1) 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)} and

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:

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 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{3\pi}{2}} .

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 2\cos(\theta)+1=0} are angles that satisfy 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)=\frac{-1}{2}} . 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)} , the

solutions 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{2\pi}{3}} 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 \theta=\frac{4\pi}{3}} .

Final Answer:
The solutions 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 \frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}} .

@@ Line 1: / Line 1: @@
 ''' 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>
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+!Foundations:
+|-
+|1) Which trigonometric identities are useful in this problem?
+|-
+|Answer:
+|-
+|1) <math>\sin^2(\theta)=1-\cos^2(\theta)</math> and
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Step 1:
+|-
+| 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
+|-
+|<math>(1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) </math>.
+|}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Step 2:
 |-
-|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>
+| 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:
 |-
-|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.
+| 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:
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+| The solutions to <math>\cos(\theta)=0</math> in <math> [0, 2\pi)</math> are <math>\theta=\frac{\pi}{2}</math> or <math>\theta=\frac{3\pi}{2}</math>.
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Step 5:
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+| The solutions to <math>2\cos(\theta)+1=0</math> are angles that satisfy <math>\cos(\theta)=\frac{-1}{2}</math>. In <math> [0, 2\pi)</math>, the
 |-
-|e) True.
+| solutions are <math>\theta=\frac{2\pi}{3}</math> or <math>\theta=\frac{4\pi}{3}</math>.
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Final Answer:
 |-
-|f) False.
+| The solutions are <math>\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}</math>.
 |}

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

Latest revision as of 20:43, 21 May 2015

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