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  • Trigonometry
  • Trigonometric identities test
  • Product to sum formulas

Product to sum formulas

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  • Question 1:
    1 pts
    		Find the correct formula and express cos⁡u⋅sin⁡v\cos u\cdot \sin v as a sum or difference of trigonometric functions
    12[cos⁡(u+v)−sin⁡(u−v)]\dfrac{1}{2}[\cos (u+v)-\sin(u-v)]
    12[sin⁡(u+v)−sin⁡(u−v)]\dfrac{1}{2}[\sin (u+v)-\sin(u-v)]
    12[sin⁡(u+v)−cos⁡(u−v)]\dfrac{1}{2}[\sin (u+v)-\cos(u-v)]
  • Question 2:
    1 pts
    		Express sin⁡3x⋅sin⁡5x\sin 3x\cdot \sin 5x as a sum or difference of trigonometric functions.
    sin⁡3x⋅sin⁡5x=12cos⁡2x−12cos⁡8x\sin 3x\cdot \sin 5x = \dfrac{1}{2}\cos 2x-\dfrac{1}{2}\cos8x
    sin⁡3x⋅sin⁡5x=12sin⁡2x−12cos⁡8x\sin 3x\cdot \sin 5x = \dfrac{1}{2}\sin 2x-\dfrac{1}{2}\cos8x
    sin⁡3x⋅sin⁡5x=12sin⁡2x−12sin⁡8x\sin 3x\cdot \sin 5x = \dfrac{1}{2}\sin 2x-\dfrac{1}{2}\sin8x
  • Question 3:
    1 pts
    		Express sin⁡(α−β)⋅cos⁡(α+β)\sin(\alpha-\beta)\cdot \cos(\alpha+\beta) as a sum or difference of trigonometric functions.

    12cos⁡2α+12cos⁡2β\dfrac{1}{2}\cos 2\alpha+\dfrac{1}{2}\cos 2\beta

    12cos⁡2α−12sin⁡2β\dfrac{1}{2}\cos 2\alpha-\dfrac{1}{2}\sin 2\beta

    12sin⁡2α+12sin⁡2β\dfrac{1}{2}\sin 2\alpha+\dfrac{1}{2}\sin 2\beta

    12sin⁡2α−12sin⁡2β\dfrac{1}{2}\sin 2\alpha-\dfrac{1}{2}\sin 2\beta

  • Question 4:
    2 pts
    		sin⁡(60∘−α)⋅sin⁡(60∘+α)=14(2cos⁡2α+1)\sin(60^{\circ}-\alpha)\cdot \sin(60^{\circ}+\alpha)=\dfrac{1}{4}\left(2\cos2\alpha+1\right)
  • Question 5:
    2 pts
    		Find the exact value of the expression.
    tan⁡20∘⋅tan⁡40∘⋅tan⁡80∘=\tan 20^{\circ}\cdot \tan 40^{\circ} \cdot \tan 80^{\circ}=
  • Question 6:
    2 pts
    		Express 4sin⁡(1+π6)cos⁡(1+π3)4\sin\left(1+\dfrac{\pi}{6}\right)\cos\left(1+\dfrac{\pi}{3}\right) as a sum or difference of trigonometric functions.
    cos⁡2−1\cos 2-1
    2cos⁡2−12\cos 2-1
    2cos⁡2+12\cos 2+1
    cos⁡2+1\cos 2+1
  • Question 7:
    2 pts
    		Express cos⁡x2cos⁡y2cos⁡x+y2\cos \dfrac{x}{2}\cos \dfrac{y}{2}\cos \dfrac{x+y}{2} as a sum or difference of trigonometric functions.
    14+14cos⁡x+14cos⁡y+14cos⁡(x+y)\dfrac{1}{4}+\dfrac{1}{4}\cos x+\dfrac{1}{4} \cos y+ \dfrac{1}{4}\cos (x+y)
    14cos⁡x+14cos⁡y+14cos⁡(x+y)\dfrac{1}{4}\cos x+\dfrac{1}{4} \cos y+ \dfrac{1}{4}\cos (x+y)
    14+14cos⁡x+14cos⁡y\dfrac{1}{4}+\dfrac{1}{4}\cos x+\dfrac{1}{4} \cos y
    14+14cos⁡(x+y)\dfrac{1}{4}+ \dfrac{1}{4}\cos (x+y)
  • Question 8:
    3 pts
    		Simplify the expression using product to sum formulas.
    sin⁡20∘sin⁡40∘sin⁡60∘sin⁡80∘=\sin20^{\circ}\sin 40^{\circ}\sin 60^{\circ}\sin 80^{\circ}=
  • Question 9:
    3 pts
    		NN is an integer in the range [0,180][0,180] such that 2cos⁡43∘cos⁡26∘=cos⁡69∘+cos⁡N.2\cos43^{\circ}\cos26^{\circ}= \cos69^{\circ}+\cos N. What is the value of NN?
    cos⁡20∘\cos 20^{\circ}
    cos⁡17∘\cos 17^{\circ}
    cos⁡27∘\cos 27^{\circ}
  • Question 10:
    3 pts
    		Express 4sin⁡αsin⁡2αsin⁡4α4\sin \alpha \sin 2\alpha \sin 4\alpha as a sum or difference of trigonometric functions.

    sin⁡2α+sin⁡4α+sin⁡6α\sin 2\alpha+\sin 4\alpha+\sin 6\alpha

    sin⁡2α+sin⁡4α−sin⁡6α\sin 2\alpha+\sin 4\alpha-\sin 6\alpha

    sin⁡2α−sin⁡4α−sin⁡6α\sin 2\alpha-\sin 4\alpha-\sin 6\alpha

    sin⁡3α−sin⁡6α−sin⁡9α\sin 3\alpha-\sin 6\alpha-\sin 9\alpha

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