Trigonometric problem? This may be really easy but is giving a hard time. I'm a beginner at trigonometry so hope I can understand it.


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*If $\sin{\frac{x}{2}}+\cos{\frac{x}{2}}= 1.4$, then $\sin{x}=?$

*If $\sin{a}-\cos{a}=a$, then $\sin(2a)=?$


The trick with these questions is finding the way to start.  We'll need two identities
$$\sin^2 t + \cos^2 t = 1, \qquad\text{and}\qquad \sin(2t) = 2\sin t \cos t$$
All we have to do is square each side of the equation
$$\left(\sin \frac{x}{2} + \cos \frac{x}{2}\right)^2 = (1.4)^2$$
A bit of algebra gives
$$\sin^2 \frac{x}{2} + \cos^2\frac{x}{2} + 2 \sin \frac{x}{2} \cos \frac{x}{2} = 1.96$$
$$1 + 2 \sin \frac{x}{2} \cos \frac{x}{2} = 1.96$$
$$2 \sin \frac{x}{2} \cos \frac{x}{2} = 0.96$$
And with the second identity, 
$$\sin\left( 2\times\frac{x}{2}\right) = 0.96$$
Thus
$$\sin x = 0.96$$
The same principle applies to the second problem
