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1. Let $$A$$ be an angle such that $$\tan(A)=\dfrac{1}{2}$$. Find the value of $$\tan(A+120^{\circ})$$, giving your answer in simplest radical form.

2. $$\cos \left(\frac\pi6 - \theta\right)$$ can be expressed in the form $$A\sin \theta + B\cos\theta$$ where $$A$$ and $$B$$ are constants. What is the value of $$\frac BA$$?

3. Consider angles $$x$$ and $$y$$ such that $$0 \le y \le x \le \frac\pi2$$ and $$\sin(x+y) = 0.9$$ while $$\sin(x-y) = 0.6$$. What is the value of $$(\sin x + \cos x)(\sin y + \cos y)$$?

4. Suppose $$\sin a + \cos b = -0.2$$ while $$\sin b + \cos a = 1.6$$. What is the value of $$\sin(a+b)$$?

Any help on these would be greatly appreciated! Thanks!

Oct 19, 2019

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1. tan(A + 120) = 4 - 2*sqrt(3)

2. A = sqrt(3) and B = 4, so A/B = sqrt(3)/4.

3. (sin x + cos x)(sin y + cos y) = 4/5.

4. sin(a + b) = 3/4.

Apr 5, 2020