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Algebra Difficulty 3.2 AMC 10/12 Find the answer

Given
f(α)=sin(πα)cos(2πα)sin(α+3π2)sin(π2+α)sin(πα).f(α)= \frac {\sin(π-α)\cos(2π-α)\sin(-α+ \frac {3π}{2})}{\sin( \frac {π}{2}+\alpha )\sin(-π -\alpha )}.
(1) Simplify f(α)f(α).
(2) If αα is an angle in the third quadrant and cos(α+π3)=35\cos\left(α+ \frac {π}{3}\right) = \frac {3}{5}, find the value of f(α)f(α).

A number or a short expression. Spacing and $ signs are ignored.

Solution

(1) We have
f(α)=sin(πα)cos(2πα)sin(α+3π2)sin(π2+α)sin(πα).f(α)= \frac {\sin(π-α)\cos(2π-α)\sin(-α+ \frac {3π}{2})}{\sin( \frac {π}{2}+\alpha )\sin(-π -\alpha )}.
Using the trigonometric identities, sin(πα)=sinα\sin(π-α) = \sin α, cos(2πα)=cosα\cos(2π-α) = \cos α, sin(α+3π2)=cosα\sin(-α+ \frac {3π}{2}) = -\cos α, and sin(π2+α)=cosα\sin(\frac {π}{2}+\alpha) = \cos α, sin(πα)=sinα\sin(-π -\alpha ) = \sin α, the expression simplifies to
f(α)=sinαcosα(cosα)cosαsinα=cosα.f(α) = \frac {\sin α \cdot \cos α \cdot (-\cos α)}{\cos α \cdot \sin α} = -\cos α.

(2) Given that αα is an angle in the third quadrant, and cos(α+π3)=35>0\cos\left(α+ \frac {π}{3}\right) = \frac {3}{5} > 0, so α+π3α+ \frac {π}{3} is an angle in the fourth quadrant.
Thus,
sin(α+π3)=1cos2(α+π3)=1(35)2=45.\sin\left(α+ \frac {π}{3}\right) = - \sqrt {1-\cos^{2}\left(α+ \frac {π}{3}\right)}= - \sqrt {1-\left(\frac {3}{5}\right)^2}= - \frac {4}{5}.
Now, we simplify f(α)f(α):
f(α)=cosα=cos[(α+π3)π3]=cos(α+π3)cosπ3+sin(α+π3)sinπ3=3512+(45)32=3104310=34310. \begin{align*} f(α) &= -\cos α \\ &= -\cos\left[\left(α+ \frac {π}{3}\right) - \frac {π}{3}\right] \\ &= -\cos\left(α+ \frac {π}{3}\right)\cos \frac {π}{3} + \sin\left(α+ \frac {π}{3}\right)\sin \frac {π}{3} \\ &= -\frac {3}{5} \cdot \frac {1}{2} + (-\frac {4}{5})\cdot \frac {\sqrt {3}}{2} \\ &= -\frac {3}{10} - \frac {4\sqrt {3}}{10} \\ &= \boxed{\frac {-3 - 4\sqrt {3}}{10}}. \end{align*}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.