Question 1(ix, x & xi) Exercise 8.3

Solutions of Question 1(ix, x & xi) of Exercise 8.3 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Use the product-to-sum formula to change the sum or difference: $2 \sin 75{\circ} \sin 15{\circ}$.

Solution.

$$\begin{align*} &\quad2 \sin 75^{\circ} \sin 15^{\circ} \\ &= \cos(75^{\circ} - 15^{\circ}) - \cos(75^{\circ} + 15^{\circ}) \\ &= \cos 60^{\circ} - \cos 90^{\circ} \\ \end{align*}$$

Use the product-to-sum formula to change the sum or difference: $4 \sin \frac{u+v}{2} \cos \frac{u-v}{2}$.

Solution.

$$\begin{align*} &4 \sin \frac{u+v}{2} \cos \frac{u-v}{2} \\ &= 2 \cdot 2 \sin \frac{u+v}{2} \cos \frac{u-v}{2} \\ &= 2[\sin\left( \frac{u+v}{2} + \frac{u-v}{2} \right) + \sin\left( \frac{u+v}{2} - \frac{u-v}{2} \right)] \\ &= 2[\sin u + \sin v ] \end{align*}$$

Use the product-to-sum formula to change the sum or difference: $2 \cos \frac{2u+2v}{2}\sin \frac{2u-2v}{2}$.

Solution.

$$\begin{align*} & 2 \cos \frac{2u+2v}{2}\sin \frac{2u-2v}{2} \\ & = 2 \cos (u+v) \sin (u-v) \\ & = \sin \left(u+v+u-v \right) - \sin\left(u+v-u+v \right) \\ & = \sin 2u - \sin 2v \end{align*}$$

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