If sin B 1 over 2 show that 3 cos B - 4 cos cubed B equals 0?

Answer 1

All you need is knowledge of the sine and cosine ratios in a right angle triangle, and Pythagoras.

Using Pythagoras you can work out the third side of the triangle which means you now know the value of cos B and so can substitute that into 3 cos B - 4 cos³ B and simplify it to see what the result is; if the result is 0, you have shown the required value.

Try working it out before reading any further.

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If you can't work it out by your self, read on:

The sine ratio is opposite/hypotenuse

The cosine ratio is adjacent/hypotenuse

In a right angled triangle Pythagoras holds:

hypotenuse² = adjacent² + opposite²

→ adjacent = √(hypotenuse² - opposite²)

If sin B = 1/2 → opposite = 1, hypotenuse = 2

→ adjacent = √(2² -1²) = √(4 - 1) = √3

→ cos B = adjacent/hypotenuse = (√3)/2

→ 3 cos B - 4 cos³ B = 3 × (√3)/2 - 4 × ((√3)/2)³

= 3(√3)/2 - 4((√3)²/2³)

= 3(√3)/2 - 4(3(√3)/8)

= 3(√3)/2 - 3(√3) × 4/8

= 3(√3)/2 - 3(√3)/2

= 0