Work out 1 2/3 ÷ 2 3/4 ?

Mathematics

1 answer:

nata0808 [166]3 years ago

6 0

1 2/3 = 5/3
2 3/4 = 11/4
5/3 ÷ 11/4 = 5/3 × 4/11
= 20/33

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Answer: see proof below

Step-by-step explanation:

Use the Double Angle Identity: cos 2A = 2cos²A - 1

Use the Difference Identity: cos (A - B) = cosA · cosB + sinA · sinB

Use the Unit Circle to evaluate: cos (π/2) = 0 & sin (π/2) = 1

Proof LHS → RHS

$\text{Given:}\qquad \qquad \qquad \qquad 2\cos^2\bigg(\dfrac{\pi}{4}-\dfrac{A}{2}\bigg)-1\\\\\\\text{Double Angle Identity:}\quad \cos2\bigg(\dfrac{\pi}{4}-\dfrac{A}{2}\bigg)\\\\\\\text{Simplify:}\qquad \qquad \qquad \quad \cos\bigg(\dfrac{\pi}{2}-A\bigg)\\\\\\\text{Difference Identity:}\qquad \cos\dfrac{\pi}{2}\cdot \cos A+\sin \dfrac{\pi}{2}\cdot \sin A\\\\\\\text{Unit Circle:}\qquad \qquad \qquad 0\cdot \cos A+1\cdot \sin A\\\\\\\text{Simplify:}\qquad \qquad \qquad \qquad \sin A$