1MA 01. TRIGONOMETRIA/1MA.01.4 FòrmulesTrigonomètriques 1MA.01.4.10DT FÒRMULES TRIGONOMÈTRIQUES

Relació fonamental

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sin«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»§#945;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»cos«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»§#945;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/math»

Se'n dedueix que:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»sin§#945;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#177;«/mo»«msqrt mathcolor=¨#003300¨»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»-«/mo»«msup»«mi mathvariant=¨bold¨»cos«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»§#945;«/mi»«/msqrt»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»cos§#945;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#177;«/mo»«msqrt mathcolor=¨#003300¨»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»-«/mo»«msup»«mi mathvariant=¨bold¨»sin«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»§#945;«/mi»«/msqrt»«/mrow»«/mstyle»«/math»

Calculadora: si cosα = 0,5 cal escriure: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt/»«mfenced»«mrow»«mn»1«/mn»«mo»-«/mo»«mn»0«/mn»«mo».«/mo»«msup»«mn»5«/mn»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/math»

Es recorda que:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»tg§#945;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»sin§#945;«/mi»«mi mathvariant=¨bold¨»cos§#945;«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.01.4.21Q 2000S Tg i sin2a a partir de sina D'un angle a del #Q quadrant, coneixeu que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfenced mathcolor=¨#003300¨ open=¨|¨ close=¨|¨»«mi mathvariant=¨bold¨»sina«/mi»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/mi»«/mrow»«/mstyle»«/math». Calculeu el valor exacte de:

a) tg a

b) sin(2a)

Es recorda la fórmula: sin (2a) = 2·sina·cosa

]]> 1.0000000 0.5000000 0 0 a) =#sol1b) =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mi>s2</mi><mn>10</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi><mo>=</mo><mfenced><mrow><mi>asin</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mfenced><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>"</mo><mi>primer</mi><mo>"</mo><mo>,</mo><mo> </mo><mo>"</mo><mi>segon</mi><mo>"</mo><mo>,</mo><mo>"</mo><mi>tercer</mi><mo>"</mo><mo>,</mo><mo>"</mo><mi>quart</mi><mo>"</mo></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c1</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>s</mi><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol11</mi><mo>=</mo><mfrac><mi>s</mi><mi>c1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>s12</mi><mo>*</mo><mi>c12</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>Q</mi><mo>=</mo><mo>"</mo><mi>primer</mi><mo>"</mo></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>sol11</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>c12</mi><mo>=</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>s12</mi><mo>=</mo><mi>s</mi></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>Q</mi><mo>=</mo><mo>"</mo><mi>segon</mi><mo>"</mo></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mo>-</mo><mi>sol11</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mn>180</mn><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>c12</mi><mo>=</mo><mo>-</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>s12</mi><mo>=</mo><mi>s</mi></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>Q</mi><mo>=</mo><mo>"</mo><mi>tercer</mi><mo>"</mo></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>sol11</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mn>180</mn><mo>+</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>c12</mi><mo>=</mo><mo>-</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>s12</mi><mo>=</mo><mo>-</mo><mi>s</mi></mtd></mtr></mtable><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mo>-</mo><mi>sol11</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mn>360</mn><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>c12</mi><mo>=</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>s12</mi><mo>=</mo><mo>-</mo><mi>s</mi></mtd></mtr></mtable></apply></apply></apply></math></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>s</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>quart</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>53.13</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>24</mn><mn>25</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>a</mi><mo>)</mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></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="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal aplicar que:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»cos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#177;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#8201;«/mo»«msqrt mathcolor=¨#000066¨»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»-«/mo»«msup»«mi mathvariant=¨bold¨»sin«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»a«/mi»«/msqrt»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»tg«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfrac mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»sina«/mi»«mi mathvariant=¨bold¨»cosa«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«/math»

i fixar-se en el quadrant per determinar els signes.

]]> 1MA.01.4.51Q Cosinus de l'angle diferència A partir de l'expressió: cos(a+b) = cosa·cosb - sina·sinb,

i pensant que cos(a-b) = cos [a+(-b)]

Determineu l'expressió de:

cos(a-b) = cos{#1}·cos{#2} {#3}sin{#4}·sin{#5}

Format: pel signe, escriu (amb lletres): més o menys

]]> 5.0000000 0.3333333 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></math></input></command></group></library></session>]]></wirisCasSession><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="casSession"/></localData></question> 1MA.01.4.52Q Sinus de l'angle suma A partir de l'expressió: cos(a-b) = cosa·cosb + sina·sinb,

i pensant que sin(a+b) = cos [90º-(a+b)]

Determineu l'expressió de:

sin(a+b) = sin{#1}·cos{#2} {#3}cos{#4}·sin{#5}

Format: pel signe, escriu (amb lletres): més o menys

]]> 5.0000000 0.3333333 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></math></input></command></group></library></session>]]></wirisCasSession><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="casSession"/></localData></question> 1MA.01.4.53Q Sinus de l'angle diferència A partir de l'expressió: sin(a+b) = sina·cosb + cosa·sinb,

i pensant que sin(a-b) = sin [a+(-b)]

Determineu l'expressió de:

sin(a-b) = sin{#1}·cos{#2} {#3}cos{#4}·sin{#5}

Format: pel signe, escriu (amb lletres): més o menys