Question

given that sin Ɵ = 2/55 and cos Φ = -2/5 with Ɵ and Φ in Q-II.

1. Determine cos Ɵ?

2. Find sin Φ?

3. What is sin (Ɵ+Φ)?

4. Find cos (Ɵ+Φ)?

5. Determine tan (Ɵ-Φ)?

6. Equivalent to sin (Ɵ-Φ)?

Answer 1

$\sin \theta = \frac{2}{55}$ and $\cos \varphi = -\frac{2}{5}$ and $\theta$ and $\varphi$-lies in the second quadrant.

1. Determine $\cos \theta$?

$$\cos^2 \theta + \sin^2 \theta = 1$$ $$\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$$

If $\theta$ lie in the second quadrant than $\cos \theta < 0$

$$\cos \theta = - \sqrt{1 - \left(\frac{2}{55}\right)^2} = - \sqrt{1 - \frac{4}{3025}} = - \frac{\sqrt{3021}}{55}$$

2. Find $\sin \varphi$

$$\cos^2 \varphi + \sin^2 \varphi = 1$$ $$\sin \varphi = \pm \sqrt{1 - \cos^2 \varphi}$$

If $\varphi$ lie in the second quadrant than $\sin \varphi > 0$

$$\sin \varphi = \sqrt{1 - \cos^2 \varphi} = \sqrt{1 - \left(-\frac{2}{5}\right)^2} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$$

3. What $\sin (\varphi + \theta)$?

$$\sin (\varphi + \theta) = \sin \varphi \cos \theta + \cos \varphi \sin \theta$$ $$\sin (\varphi + \theta) = \frac{\sqrt{21}}{5} \left(- \frac{\sqrt{3021}}{55}\right) + \left(- \frac{2}{5}\right) \frac{2}{55} = - \frac{3 \sqrt{7049} + 4}{275}$$

4. Find $\cos (\varphi + \theta)$?

$$\cos (\varphi + \theta) = \left(- \frac{2}{5}\right) \left(- \frac{\sqrt{3021}}{55}\right) - \frac{\sqrt{21}}{5} \frac{2}{55} = \frac{2 \sqrt{3021} - 2 \sqrt{21}}{275}$$

5. Determine $\operatorname{tg}(\theta - \varphi)$

$$\operatorname{tg}(\theta - \varphi) = \frac{\operatorname{tg} \theta + \operatorname{tg} \varphi}{1 - \operatorname{tg} \theta \operatorname{tg} \varphi}$$ $$tg\theta = \frac{\sin\theta}{\cos\theta} = \frac{\frac{2}{55}}{-\frac{\sqrt{3021}}{55}} = -\frac{2}{\sqrt{3021}}$$ $$tg\varphi = \frac{\sin\varphi}{\cos\varphi} = \frac{\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = -\frac{\sqrt{21}}{2}$$ $$\tg(\theta - \varphi) = \frac{-\frac{2}{\sqrt{3021}} + -\frac{\sqrt{21}}{2}}{1 - \frac{2}{\sqrt{3021}} \cdot \frac{\sqrt{21}}{2}} = \frac{-4 - \sqrt{63441}}{2\sqrt{3021} - 2\sqrt{21}}$$

6. Equivalent to $\sin(\theta - \varphi)$?

$$\sin(\theta - \varphi) = \sin\theta\cos\varphi - \cos\theta\sin\varphi$$ $$\sin(\theta - \varphi) = \frac{2}{55}\left(-\frac{2}{5}\right) + \frac{\sqrt{3021}}{55} \frac{\sqrt{21}}{5} = \frac{-4 + 3\sqrt{7049}}{275}$$