Question

assignment 7.5: solving trigonometric equations
score: 70/100 answered: 8/10
question 9
solve ( 7 cos (2 phi)=7 sin ^{2}(phi)+5 ) for all solutions ( 0 leq phi<2 pi ).
( phi= )
give your answers accurate to at least 2 decimal places and in a list separated by commas.
question help: video ( square ) post to forum

Explanation:

Step1: Use double - angle formula

Recall the double - angle formula \(\cos(2\phi)=1 - 2\sin^{2}\phi\). The equation \(7\cos(2\phi)=7\sin^{2}\phi + 5\) becomes \(7(1 - 2\sin^{2}\phi)=7\sin^{2}\phi+5\).

Step2: Expand and simplify the equation

Expand the left - hand side: \(7-14\sin^{2}\phi=7\sin^{2}\phi + 5\).
Move all terms involving \(\sin^{2}\phi\) to one side: \(-14\sin^{2}\phi-7\sin^{2}\phi=5 - 7\).
Combine like terms: \(-21\sin^{2}\phi=-2\).
So, \(\sin^{2}\phi=\frac{2}{21}\), and \(\sin\phi=\pm\sqrt{\frac{2}{21}}\approx\pm0.31\).

Step3: Find the values of \(\phi\)

When \(\sin\phi = 0.31\), \(\phi=\sin^{-1}(0.31)\approx0.31\) or \(\phi=\pi-\sin^{-1}(0.31)\approx2.83\).
When \(\sin\phi=-0.31\), \(\phi=\pi+\sin^{-1}(0.31)\approx3.45\) or \(\phi = 2\pi-\sin^{-1}(0.31)\approx5.97\).

Answer:

\(0.31,2.83,3.45,5.97\)