Question

the value of 16sin^3teta+ 8 cos^3 teta is

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Answer 1

answer

16 \sin^3 \theta+ 8\cos^3 \theta =16sin

3

θ+8cos

3

θ=

To prove:

Simplify.

Solution:

Formula:

$$\begin{lgathered}\bold{\sin 3\theta= 3\sin \theta -4\sin^3 \theta}\\\end{lgathered}$$

$$\begin{lgathered}\bold{4\sin^3 \theta= 3\sin \theta -\sin 3\theta}\\\end{lgathered}$$

$$\bold{ \cos3\theta= 4\cos^3 \theta-3\cos \theta}$$

$$\bold{ 4\cos^3 \theta=\cos3\theta +3\cos \theta}$$

$$\Rightarrow 16 \sin^3 \theta+ 8\cos^3 \theta = 4 \cdot 4 \sin^3 \theta+ 2 \cdot 4\cos^3 \theta$$

$$\begin{lgathered}= 4 (3\sin \theta- \sin 3\theta)+ 2 (\cos 3\theta+3\cos \theta)\\\\= 12\sin \theta- 4\sin 3\theta+ 2\cos 3\theta+6\cos \theta\end{lgathered}$$

The final answer is= $$\begin{lgathered}12\sin \theta- 4\sin 3\theta+ 2\cos 3\theta+6\cos \theta \\\\\end{lgathered}$$