Home fsc 1st year math solutions Trigonometric Ratios of Allied Angles Exercise 10.1

Trigonometric Ratios of Allied Angles Exercise 10.1

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Trigonometric Identities 10.1
Trigonometric Identities 10.1
  1. Without using the tables, find the values of:
    i) \sin \left(-780^{\circ}\right)
    ii) \cot \left(-855^{\circ}\right)
    iii) \csc \left(2040^{\circ}\right)
    iv) \sec \left(-960^{\prime}\right)
    v) \tan \left(1110^{\circ}\right)
    vi) \sin (-300)
  2. Express each of the following as a trigonometric function of an angle of positive degre measure of less than 45^{\circ}.
    i) \sin 196^{\circ}
    ii) \cos 147^{\circ}
    iii) \sin 319^{\prime}
    iv) \cos 254
    v) \tan 294
    vi) \cos 728
    vii) \sin \left(-625^{\circ}\right) \quad viii) \cos \left(-435^{\circ}\right) \quad ix) \sin 150^{\circ}
  3. Prove the following:
    i) \sin \left(180^{\circ}+\alpha\right) \sin (90-\alpha)=\sin \alpha \cos \alpha
    ii) \sin 780^{\circ} \sin 480^{\circ}+\cos 120^{\circ} \sin 30^{\circ}=\frac{1}{2}
    iii) \cos 306^{\prime}+\cos 234^{\prime}+\cos 162^{\prime}+\cos 18^{\prime}=0
    Iv) \cos 330^{\circ} \sin 600^{\circ}+\cos 120^{\circ} \sin 150^{\circ}=1
  1. Prove that:
    i) \sin ^2(\pi+\theta) \tan \left(\frac{3 \pi}{2}+\theta\right) \frac{2}{\cot ^2\left(\frac{3 \pi}{2}-\theta\right) \cos ^2(\pi-\theta) \operatorname{coses}(2 \pi-\theta)}=\cos \theta
    ii) \frac{\cos \left(90^{\circ}+\theta\right) \sec (-\theta) \tan \left(180^{\circ}-\theta\right)}{\sec \left(360^{\circ}-\theta\right) \sin \left(180^{\circ}+\theta\right) \cot \left(90^{\circ}-\theta\right)}=-1
  2. If \alpha, \beta, \gamma are the angles of a triangle \mathrm{ABC}, then prove that
    i) \sin (\alpha+\beta)=\sin \gamma
    ii) \cos \left(\frac{\alpha+\beta}{2}\right)=\sin \frac{\gamma}{2}
    iii) \cos (\alpha+\beta)=\cos \gamma
    Iv) \tan (\alpha+\beta)+\tan \gamma=0
Previous articleApplications of basic Identities Exercise 10.2
Next articleSum, Difference and Product of Sines and Cosines Exercise 10.4
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