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Chapter 1: Problem 33
Find the number(s) \(x\) in the interval \([0.2 \pi]\) which satisfy the equation. $$\tan x / 2=1$$.
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Given that \(f\) is defined for all real numbers, show that the function \(h(x)=f(x)-f(-x)\) is an odd function.
Determine the range of \(y=\frac{2 x}{4-x}\) (a) by writing \(y\) in the form \(a+\frac{b}{4-x}\) (b) by first solving the equation for \(x\)
Evaluate. \(\frac{8 !}{3 ! 5 !}\).
Suppose that \(l_{1}\) and \(l_{2}\) are two nonvertical lines. If \(m_{1} m_{3}=\) \(-1,\) then \(l_{1}\) and \(l_{2}\) intersect at right angles. Show that if \(l_{1}\) and \(l_{2}\) do not interscet al right angles, then the angle \(\alpha\) between \(l_{1}\) and \(l_{2}\) (see Scction 1.4 ) is given by the formula $$\tan \alpha=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$$. HINT: Derive the identity $$\tan \left(\theta_{1}-\theta_{2}\right)=\frac{\tan \theta_{1}-\tan \theta_{2}}{1+\tan \theta_{1} \tan \theta_{2}}$$ by expressing the right side in terms of sines and cosines.
Verify the following identities: $$\sin \left(\frac{1}{2} \pi-\theta\right)=\cos \theta, \quad \cos \left(\frac{1}{2} \pi-\theta\right)=\sin \theta$$.
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