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Homefsc 1st year math solutionsApplications of basic Identities Exercise 10.2

Applications of basic Identities Exercise 10.2

  1. Prove that
    i) \sin \left(180^{\circ}+\theta\right)=-\sin \theta
    ii) \cos \left(180^{\circ}+\theta\right)=-\cos \theta
    iii) \tan \left(270^{\circ}-\theta\right)=\cot \theta
    iv) \cos \left(\theta-18 \theta^{\circ}\right)=\cos \theta
    v) \cos (270+\theta)=\sin \theta
    vi) \sin \left(\theta+27 \theta^{\prime}\right)=\cos \theta
    vii) \tan \left(180^{\circ}+\theta\right)=\tan \theta
    viii) \cos \left(360^{\circ}-\theta\right)=\cos \theta
  2. Find the values of the following:
    i) \sin 15^{\circ}
    ii) \cos 15
    iii) \tan 15
    \begin{array}{ll}\text { iv) } \sin 105^{\circ} & \text { v) } \cos 105^{\circ}\end{array}
    vi) \tan 105
    (Hint: 15^{\prime}=\left(\begin{array}{ll}45^{\prime} & 30^{\prime}\end{array}\right) and -105^{\prime} \quad\left(60^{\prime} \quad 45^{\prime}\right) ):
  3. Prove that:
    i) \sin \left(45^{\circ}+\alpha\right)=\frac{1}{\sqrt{2}}(\sin \alpha+\cos \alpha)
    ii) \cos \left(\alpha+45^{\circ}\right)=\frac{1}{\sqrt{2}}(\cos \alpha-\sin \alpha)
  4. Prove that:
    i) \tan \left(45^{\circ}+A\right) \tan \left(45^{\circ}-A\right)=1
    ii) \tan \left(\frac{\pi}{4}-\theta\right)+\tan \left(\frac{3 \pi}{4}+\theta\right)=0
    iii) \sin \left(\theta+\frac{\pi}{6}\right)+\cos \left(\theta+\frac{\pi}{3}\right)=\cos \theta
    iv) \frac{\sin \theta-\cos \theta \tan \frac{\theta}{2}}{\cos \theta+\sin \theta \tan \frac{\theta}{2}}=\tan \frac{\theta}{2}
  5. v) \frac{1-\tan \theta \tan \phi}{1+\tan \theta \tan \phi}=\frac{\cos (\theta+\phi)}{\cos (\theta-\phi)}
  6. Show that: \cos (\alpha+\beta) \cos (\alpha-\beta)=\cos ^2 \alpha-\sin ^2 \beta=\cos ^2 \beta-\sin ^2 \alpha
  7. Show that: \frac{\sin (\alpha+\beta)+\sin (\alpha-\beta)}{\cos (\alpha+\beta)+\cos (\alpha-\beta)}=\tan \alpha
  8. Show that:
    i) \cot (\alpha+\beta)=\frac{\cot \alpha \cot \beta-1}{\cot \alpha+\cot \beta}
    ii) \cot (\alpha-\beta)=\frac{\cot \alpha \cot \beta+1}{\cot \beta-\cot \alpha}
    iii) \frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta}=\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}
  9. If \sin \alpha=\frac{4}{5} and \cos \beta \frac{40}{41}, where 0<\alpha<\frac{\pi}{2} and 0<\beta<\frac{\pi}{2}.
    Show that \sin (\alpha-\beta)=\frac{133}{205}
  10. If \sin \alpha=\frac{4}{5}< and \sin \beta=\frac{12}{13} where \frac{\pi}{2} \quad \alpha \quad \pi and \frac{\pi}{2} \quad \beta \quad \pi. Find
    i) \sin (\alpha+\beta)
    ii) \cos (\alpha+\beta)
    iii) \tan (\alpha+\beta)
    iv) \sin (\alpha-\beta)
    v) \cos (\alpha-\beta)
    vi) \tan (\alpha-\beta)
    In which quadrants do the terminal sides of the angles of measures (\alpha+\beta) and (\alpha-\beta) lie?
  11. Find \sin (\alpha+\beta) and \cos (\alpha+\beta), given that
    i) \tan \alpha=\frac{3}{4}, \cos \beta=\frac{5}{13} and neither the terminal side of the angle of measure \alpha nor that of \beta is in the I quadrant.
  12. ii) \tan \alpha=\frac{15}{8} and \sin A \quad \frac{7}{25} and neither the terminal side of the angle of measure \alpha nor that of \beta is in the Iv quadrant.
  13. Prove that \frac{\cos 8^{\circ} \sin 8^{\circ}}{\cos 8^{\circ} \sin 8^{\circ}} \tan 37
  14. If \alpha, \beta, \gamma are the angles of a triangle A B C, show that \cot \frac{\alpha}{2}+\cot \frac{\beta}{2}+\cot \frac{\gamma}{2}=\cot \frac{\alpha}{2} \cot \frac{\beta}{2} \cot \frac{\gamma}{2}
  15. If \alpha+\beta+\gamma=180^{\circ}, show that

        \[\cot \alpha \cot \beta+\cot \beta \cot \gamma+\cot \gamma \cot \alpha=1\]

  16. Express the following in the form \mathrm{r} \sin (\theta+\phi) or \mathrm{r} \sin (\theta-\phi), where terminal sides of the angles of measures \theta and \phi are in the first quadrant:
    i) 12 \sin \theta+5 \cos \theta
    ii) 3 \sin \theta-4 \cos \theta
    iii) \sin \theta-\cos \theta
    iv) 5 \sin \theta-4 \cos \theta
    v) \sin \theta+\cos \theta.
    vi) 3 \sin \theta-5 \cos \theta
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