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7.2.2E: Identidades de suma y resta (ejercicios)

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    Sección 7.2 Ejercicios

    Encuentra un valor exacto para cada uno de los siguientes.

    1. \(\sin \left(75{}^\circ \right)\)

    2. \(\sin \left(195{}^\circ \right)\)

    3. \({\rm cos}(165{}^\circ )\)

    4. \({\rm cos}(345{}^\circ )\)

    5. \(\cos \left(\dfrac{7\pi }{12} \right)\)

    6. \(\cos \left(\dfrac{\pi }{12} \right)\)

    7. \(\sin \left(\dfrac{5\pi }{12} \right)\)

    8. \(\sin \left(\dfrac{11\pi }{12} \right)\)

    Reescribir en términos de\(\sin \left(x\right)\) y\(\cos \left(x\right)\).

    9. \(\sin \left(x+\dfrac{11\pi }{6} \right)\)

    10. \(\sin \left(x-\dfrac{3\pi }{4} \right)\)

    11. \(\cos \left(x-\dfrac{5\pi }{6} \right)\)

    12. \(\cos \left(x+\dfrac{2\pi }{3} \right)\)

    Simplifica cada expresión.

    13. \(\csc \left(\dfrac{\pi }{2} -\; t\right)\)

    14. \(\sec \left(\dfrac{\pi }{2} -w\right)\)

    15. \(\cot \left(\dfrac{\pi }{2} -x\right)\)

    16. \(\tan \left(\dfrac{\pi }{2} -x\right)\)

    Reescribir el producto como una suma.

    17. \(16\sin \left(16x\right)\sin \left(11x\right)\)

    18. \(20\cos \left(36t\right)\cos \left(6t\right)\)

    19. \(2\sin \left(5x\right)\cos \left(3x\right)\)

    20. \(10\cos \left(5x\right)\sin \left(10x\right)\)

    Reescribir la suma como un producto.

    21. \(\cos \left(6t\right)+\cos \left(4t\right)\)

    22. \(\cos \left(6u\right)+\cos \left(4u\right)\)

    23. \(\sin \left(3x\right)+\sin \left(7x\right)\)

    24. \(\sin \left(h\right)+\sin \left(3h\right)\)

    25. Dado\(\sin \left(a\right)=\dfrac{2}{3}\) y\(\cos \left(b\right)=-\dfrac{1}{4}\), con\(a\) y\(b\) ambos en el intervalo\(\left[\dfrac{\pi }{2} ,\pi \right)\):

    a. Encontrar\(\sin \left(a+b\right)\)
    b. Buscar\(\cos \left(a-b\right)\)

    26. Dado\(\sin \left(a\right)=\dfrac{4}{5}\) y\(\cos \left(b\right)=\dfrac{1}{3}\), con\(a\) y\(b\) ambos en el intervalo\(\left[0,\dfrac{\pi }{2} \right)\):

    a. Encontrar\(\sin \left(a-b\right)\)
    b. Buscar\(\cos \left(a+b\right)\)

    Resuelve cada ecuación para todas las soluciones.

    27. \(\sin \left(3x\right)\cos \left(6x\right)-\cos \left(3x\right)\sin \left(6x\right)= -0.9\)

    28. \(\sin \left(6x\right)\cos \left(11x\right)-\cos \left(6x\right)\sin \left(11x\right)= -0.1\)

    29. \(\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)=1\)

    30. \(\cos \left(5x\right)\cos \left(3x\right)-\sin \left(5x\right)\sin \left(3x\right)=\dfrac{\sqrt{3} }{2}\)

    Resuelve cada ecuación para todas las soluciones.

    31. \(\cos \left(5x\right)=-\cos \left(2x\right)\)

    32. \(\sin \left(5x\right)=\sin \left(3x\right)\)

    33. \(\cos \left(6\theta \right)-\cos \left(2\theta \right)=\sin \left(4\theta \right)\)

    34. \(\cos \left(8\theta \right)-\cos \left(2\theta \right)=\sin \left(5\theta \right)\)

    Reescribir como una sola función del formulario\(A\sin (Bx+C)\).

    35. \(4\sin \left(x\right)-6\cos \left(x\right)\)

    36. \(-\sin \left(x\right)-5\cos \left(x\right)\)

    37. \(5\sin \left(3x\right)+2\cos \left(3x\right)\)

    38. \(-3\sin \left(5x\right)+4\cos \left(5x\right)\)

    Resuelve las dos primeras soluciones positivas.

    39. \(-5\sin \left(x\right)+3\cos \left(x\right)=1\)

    40. \(3\sin \left(x\right)+\cos \left(x\right)=2\)

    41. \(3\sin \left(2x\right)-5\cos \left(2x\right)=3\)

    42. \(-3\sin \left(4x\right)-2\cos \left(4x\right)=1\)

    Simplificar.

    43. \(\dfrac{\sin \left(7t\right)+\sin \left(5t\right)}{\cos \left(7t\right)+\cos \left(5t\right)}\)

    44. \(\dfrac{\sin \left(9t\right)-\sin \left(3t\right)}{\cos \left(9t\right)+\cos \left(3t\right)}\)

    Demostrar la identidad.

    44. \(\tan \left(x+\dfrac{\pi }{4} \right)=\dfrac{\tan \left(x\right)+1}{1-\tan \left(x\right)}\)

    45. \(\tan \left(\dfrac{\pi }{4} -t\right)=\dfrac{1-\tan \left(t\right)}{1+\tan \left(t\right)}\)

    46. \(\cos \left(a+b\right)+\cos \left(a-b\right)=2\cos \left(a\right)\cos \left(b\right)\)

    47. \(\dfrac{\cos \left(a+b\right)}{\cos \left(a-b\right)} =\dfrac{1-\tan \left(a\right)\tan \left(b\right)}{1+\tan \left(a\right)\tan \left(b\right)}\)

    48. \(\dfrac{\tan \left(a+b\right)}{\tan \left(a-b\right)} =\dfrac{\sin \left(a\right)\cos \left(a\right)+\sin \left(b\right)\cos \left(b\right)}{\sin \left(a\right)\cos \left(a\right)-\sin \left(b\right)\cos \left(b\right)}\)

    49. \(2\sin \left(a+b\right)\sin \left(a-b\right)=\cos \left(2b\right)-{\rm cos}(2a)\)

    50. \(\dfrac{\sin \left(x\right)+\sin \left(y\right)}{\cos \left(x\right)+\cos \left(y\right)} =\tan \left(\dfrac{1}{2} \left(x+y\right)\right)\)

    Demostrar la identidad.

    51. \(\dfrac{\cos \left(a+b\right)}{\cos \left(a\right)\cos \left(b\right)} =1-\tan \left(a\right)\tan \left(b\right)\)

    52. \(\cos \left(x+y\right)\cos \left(x-y\right)=\cos ^{2} x-\sin ^{2} y\)

    53. Usar las identidades de suma y diferencia para establecer la identidad de producto a suma

    \(\sin (\alpha )\sin (\beta )=\dfrac{1}{2} \left(\cos (\alpha -\beta )-\cos (\alpha +\beta )\right)\)

    54. Usar las identidades de suma y diferencia para establecer la identidad de producto a suma

    \(\cos (\alpha )\cos (\beta )=\dfrac{1}{2} \left(\cos (\alpha +\beta )+\cos (\alpha -\beta )\right)\)

    55. Usar las identidades de producto a suma para establecer la identidad de suma a producto

    \(\cos \left(u\right)+\cos \left(v\right)=2\cos \left(\dfrac{u+v}{2} \right)\cos \left(\dfrac{u-v}{2} \right)\)

    56. Usar las identidades de producto a suma para establecer la identidad de suma a producto

    \(\cos \left(u\right)-\cos \left(v\right)=-2\sin \left(\dfrac{u+v}{2} \right)\sin \left(\dfrac{u-v}{2} \right)\)

    Responder

    1. \(\dfrac{\sqrt{2} + \sqrt{6}}{4}\)

    3. \(\dfrac{-\sqrt{2} - \sqrt{6}}{4}\)

    5. \(\dfrac{\sqrt{2} - \sqrt{6}}{4}\)

    7. \(\dfrac{\sqrt{2} + \sqrt{6}}{4}\)

    9. \(\dfrac{\sqrt{3}}{2}\sin(x) - \dfrac{1}{2} \cos(x)\)

    11. \(-\dfrac{\sqrt{3}}{2}\cos(x) + \dfrac{1}{2} \sin(x)\)

    13. \(\sec(t)\)

    15. \(\tan(x)\)

    17. \(8(\cos(5x) - \cos(27x))\)

    19. \(\sin(8x) + \sin (2x)\)

    21. \(2 \cos(5t) \cos(t)\)

    23. \(2 \sin(5x) \cos(2x)\)

    25. a.\((\dfrac{2}{3})(-\dfrac{1}{4}) + (-\dfrac{\sqrt{5}}{3})(\dfrac{\sqrt{15}}{4}) = \dfrac{-2-5\sqrt{3}}{12}\)
    b.\((-\dfrac{\sqrt{5}}{3})(-\dfrac{1}{4}) + (\dfrac{2}{3})(\dfrac{\sqrt{15}}{4}) = \dfrac{\sqrt{5} + 2\sqrt{15}}{12}\)

    27. \(0.373 + \dfrac{2\pi}{3} k\)y\(0.674 + \dfrac{2\pi}{3} k\), donde\(k\) es un entero

    29. \(2 \pi k\), donde\(k\) es un número entero

    31. \(\dfrac{\pi}{7} + \dfrac{4\pi}{7} k\),\(\dfrac{3\pi}{7} + \dfrac{4\pi}{7} k\),\(\dfrac{\pi}{3} + \dfrac{4\pi}{3} k\), y\(\pi + \dfrac{4\pi}{3} k\), donde\(k\) es un entero

    33. \(\dfrac{7\pi}{12} + \pi k\),\(\dfrac{11\pi}{12} + \pi k\), y\(\dfrac{\pi}{4} k\), donde\(k\) es un entero

    35. \(2 \sqrt{13} \sin (x + 5.3004)\)o\(2\sqrt{13} \sin(x - 0.9828)\)

    37. \(\sqrt{29} \sin(3x + 0.3805)\)

    39. 0.3681, 3.8544

    41. 0.7854, 1.8158

    43. \(\tan(6t)\)

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