(a) If sin p = \(\frac{1}{2}\) and cos q = \(\frac{1}{3}\), evaluate sin(p - q), where 0\(^o\) \(\geq\) p \(\geq\) 90\(^o\) and 90\(^o\) \(\geq\) q \(\geq\)...
Question 1 Report
(a) If sin p = \(\frac{1}{2}\) and cos q = \(\frac{1}{3}\), evaluate sin(p - q), where 0\(^o\) \(\geq\) p \(\geq\) 90\(^o\) and 90\(^o\) \(\geq\) q \(\geq\) 180\(^o\)
b) Using trapezum rule with seven ordinates, evaluate \(\int^4_1\frac{2}{\sqrt{x + 3}}\)dx
Answer Details
(a) \(\sin p=\tfrac12\) with \(p\) acute gives \(p=30^\circ\), so \(\cos p=\tfrac{\sqrt3}{2}\). Since \(90^\circ (b) Trapezium rule, seven ordinates \(\Rightarrow\) six strips, \(h=\dfrac{4-1}{6}=0.5\), with \(f(x)=\dfrac{2}{\sqrt{x+3}}\): (The exact value \(\big[4\sqrt{x+3}\big]_1^4=4(\sqrt7-2)=2.58\) confirms this.)
\[\int_1^4 f\,dx\approx\frac{h}{2}\Big[(y_0+y_6)+2(y_1+y_2+y_3+y_4+y_5)\Big]=\frac{0.5}{2}\big[1.7559+2(4.2910)\big].\]
\[=0.25(1.7559+8.5820)=0.25(10.3379)\approx 2.58.\]
x 1 1.5 2 2.5 3 3.5 4 f(x) 1.0000 0.9428 0.8944 0.8528 0.8165 0.7845 0.7559
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