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Parin Sir > Quantitative Aptitude (Maths) > T-Ratios & Identities > Examples 16 to 18 of T-Ratios & Identities > Examples 16 to 18 of T-Ratios & Identities

Examples 16 to 18 of T-Ratios & Identities

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(16) If  sinθ  =  \frac{3}{5}, find the values of

         sin 2θ, cos 2θ, tan 2θ, sin 3θ  and  cos 3θ.

Soln. cosθ  =  \sqrt{1-{sin}^2{\theta}}  =  \sqrt{1-\frac{9}{25}}  =  \sqrt{\frac{16}{25}} = \frac{4}{5}.

∴  sin 2θ  =  2 sinθ cosθ  =  2 \times \frac{3}{5} \times \frac{4}{5}  =  \frac{24}{25};

cos 2θ  =   1 – 2 sin2θ   =  1 – 2 \times \left(\frac{3}{5}\right)^2=   \frac{7}{25};

tan 2θ  =  \frac{sin\ 2\theta}{cos \ 2\theta} = \frac{24}{7};

sin 3θ  =  3 sinθ – 4 sin3  =  3\times \frac{3}{5} - 4 \times \left(\frac{3}{5}\right)^3 = \frac{117}{125};

cos 3\theta  =  ( 4 cos3θ – 3 cosθ  )  =  4 \times \left(\frac{4}{5}\right)^3 - 3 \times \frac{4}{5} = -\frac{44}{125}.

 (17) If  tan \theta = \frac{3}{4}, find the values of cos 2θ  and  sin 2θ.

Soln. cos 2\theta = \frac{1-{tan}^2{\theta}}{1+{tan}^2{\theta}}  =  \frac{1-\frac{9}{16}}{1+\frac{9}{16}}  =  \frac{7}{16}\times\frac{16}{25}  =  \frac{7}{25}.

sin 2\theta = \frac{2tan{\theta}}{1+tan}^2{\theta} 

=  \frac{2\times\frac{3}{4}}{1+\frac{9}{16}}

=  \frac{3}{2}\times\frac{16}{25}

=  \frac{24}{25}.

(18) Evaluate :  cos2 75o – sin2 15o.

Soln. We know that  cos2 α – sin2 β  =  cos (α + β ). cos (α – β ).

∴ cos2 75o – sin2 15o  =  cos ( 75o + 15o). cos ( 75o – 15o)

=  cos 90o.cos 60o  =  0 \times \frac{1}{2}  =  0.

Best of Luck…

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