Trigonometría 5. Trigonometría: 3 Determinar ángulo (resolviendo seno) y coordenadas que faltan del punto Observe la siguiente figura:

 

 

Sabiendo que

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»O«/mi»«mo»=«/mo»«mo»#«/mo»«mi»O«/mi»«mn»1«/mn»«mo»§#160;«/mo»«mspace linebreak=¨newline¨/»«mi»P«/mi»«mo»=«/mo»«mo»#«/mo»«mi»P«/mi»«mn»1«/mn»«mspace linebreak=¨newline¨/»«mi»Q«/mi»«mo»=«/mo»«mo»#«/mo»«mi»Q«/mi»«mn»1«/mn»«mspace linebreak=¨newline¨/»«mi»R«/mi»«mo»=«/mo»«mo»#«/mo»«mi»R«/mi»«mn»1«/mn»«/math»

 

y además se sabe que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»O«/mi»«mi»A«/mi»«mo»=«/mo»«mi»P«/mi»«mi»B«/mi»«mo»=«/mo»«mi»Q«/mi»«mi»C«/mi»«mo»=«/mo»«mi»R«/mi»«mi»D«/mi»«mo»=«/mo»«mo»#«/mo»«mi»L«/mi»«mn»3«/mn»«/math» y que el punto «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»#«/mo»«mi»X«/mi»«mn»1«/mn»«mo»=«/mo»«mfenced»«mrow»«mo»#«/mo»«mi»x«/mi»«mn»1«/mn»«mo»,«/mo»«mi»y«/mi»«mo»,«/mo»«mi»z«/mi»«/mrow»«/mfenced»«/math» determine el valor de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#952;«/mi»«/math» y de las coordenadas restantes:

 

Obs: Ingrese el resultado de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#952;«/mi»«/math» en radianes.

 

]]> 1.0000000 0.3333333 0 0 θ=#t1y=#y1z=#z1]]> <question><wirisCasSession><![CDATA[<session lang="es" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>L1</mi><mo>=</mo><mi>aleatorio</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>L2</mi><mo>=</mo><mi>aleatorio</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>L3</mi><mo>=</mo><mi>aleatorio</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>15</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>L2</mi><mo>&gt;</mo><mi>L1</mi><mo>∧</mo><mi>L2</mi><mo>&gt;</mo><mi>L3</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O1</mi><mo>=</mo><mi>sustituir_cadena</mi><mo>(</mo><mo>&quot;</mo><mo>(</mo><mo>#</mo><mn>1</mn><mo>,</mo><mo>#</mo><mn>2</mn><mo>,</mo><mo>#</mo><mn>3</mn><mo>)</mo><mo>&quot;</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P1</mi><mo>=</mo><mi>sustituir_cadena</mi><mo>(</mo><mo>&quot;</mo><mo>(</mo><mo>#</mo><mn>1</mn><mo>,</mo><mo>#</mo><mn>2</mn><mo>,</mo><mo>#</mo><mn>3</mn><mo>)</mo><mo>&quot;</mo><mo>,</mo><mi>L1</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q1</mi><mo>=</mo><mi>sustituir_cadena</mi><mo>(</mo><mo>&quot;</mo><mo>(</mo><mo>#</mo><mn>1</mn><mo>,</mo><mo>#</mo><mn>2</mn><mo>,</mo><mo>#</mo><mn>3</mn><mo>)</mo><mo>&quot;</mo><mo>,</mo><mi>L1</mi><mo>,</mo><mi>L2</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R1</mi><mo>=</mo><mi>sustituir_cadena</mi><mo>(</mo><mo>&quot;</mo><mo>(</mo><mo>#</mo><mn>1</mn><mo>,</mo><mo>#</mo><mn>2</mn><mo>,</mo><mo>#</mo><mn>3</mn><mo>)</mo><mo>&quot;</mo><mo>,</mo><mn>0</mn><mo>,</mo><mi>L2</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>an</mi><mo>=</mo><mi>aleatorio</mi><mo>{</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>}</mo><mo>*</mo><pi/></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</mi><mo>=</mo><mi>an</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>=</mo><mi>aleatorio</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U1</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>→</mo><mi>A</mi><mo>,</mo><mn>2</mn><mo>→</mo><mi>B</mi><mo>,</mo><mn>3</mn><mo>→</mo><mi>C</mi><mo>,</mo><mn>4</mn><mo>→</mo><mi>D</mi><mo>}</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X1</mi><mo>=</mo><mi>U1</mi><mo>(</mo><mi>w1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mi>L3</mi><mo>*</mo><mi>sen</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mi>L3</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>L1</mi><mo>+</mo><mi>L3</mi><mo>*</mo><mi>sen</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mi>L3</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>L1</mi><mo>+</mo><mi>L3</mi><mo>*</mo><mi>sen</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd><mtd><mi>L2</mi></mtd><mtd><mi>L3</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi>L3</mi><mo>*</mo><mi>sen</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd><mtd><mi>L2</mi></mtd><mtd><mi>L3</mi><mo>*</mo><mi>cos</mi><mo>(</mo><mi>an</mi><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x1</mi><mo>=</mo><msub><mi>M</mi><mrow><mi>w1</mi><mo>,</mo><mn>1</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><msub><mi>M</mi><mrow><mi>w1</mi><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z1</mi><mo>=</mo><msub><mi>M</mi><mrow><mi>w1</mi><mo>,</mo><mn>3</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>theta1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>theta1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sen</mi><mo>(</mo><mfrac><pi/><mn>12</mn></mfrac><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>4</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>4</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#952;</mi><mo>=</mo><mo>#</mo><mi>t</mi><mn>1</mn><mspace linebreak="newline"/><mi>y</mi><mo>=</mo><mo>#</mo><mi>y</mi><mn>1</mn><mspace linebreak="newline"/><mi>z</mi><mo>=</mo><mo>#</mo><mi>z</mi><mn>1</mn><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question>