Roots 1

Geometry Level 3

If sin ⁡ θ \sin \theta and cos ⁡ θ \cos \theta are two roots of 25 x 2 + 5 x − 12 = 0 25x^{2}+5x-12=0 , then what is sin ⁡ 2 θ \sin 2 \theta ?
You can also try Roots 2

-.96 -1 -.97 .97 -.95 0 .96 1

Farhabi Mojib by Farhabi Mojib , 23, Bangladesh

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1 solution

Farhabi Mojib
Jun 17, 2017

If α \alpha and β \beta are two roots of 25 x 2 + 5 x − 12 = 0 25x^{2}+5x-12=0
Then by Vieta's formula α β \alpha \beta =- 12 25 \frac{12}{25}
So, s i n θ c o s θ sin \theta cos \theta = − 12 25 -\frac{12}{25}
s i n 2 θ sin 2 \theta = 2 s i n θ c o s θ 2sin \theta cos \theta
= 2 × − 12 25 2 \times -\frac{12}{25}
= − . 96 \boxed{-.96}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

More importantly, you should first answer the question of "Does there exist a θ \theta that satisfies the conditions of the problem?

IE IF the question was "If sin ⁡ θ \sin \theta and cos ⁡ θ \cos \theta are two roots of 25 x 2 − 12 = 0 25x^{2}-12=0 , then what is sin ⁡ 2 θ \sin 2 \theta ?", what would your answer be?

Calvin Lin Staff - 3 years, 11 months ago

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