109k views
5 votes
Sec(9x + 2) = csc(-x + 24)

User HeartWare
by
8.1k points

1 Answer

5 votes

Consider the given expression,


\begin{gathered} \sec (9x+2)=\csc (-x+24) \\ \sec (9x+2)=\csc (24-x) \end{gathered}

Consider the formulae,


\begin{gathered} \sec \theta=(1)/(\cos\theta) \\ \csc \theta=(1)/(\sin\theta) \\ \sin (90-\theta)=\cos \theta \end{gathered}

Then the expression is transformed as,


\begin{gathered} (1)/(\cos(9x+2))=(1)/(\sin (24-x)) \\ \sin (24-x)=\cos (9x+2) \\ \sin (24-x)=\sin \mleft\lbrace90-(9x+2)\mright\rbrace \\ \sin (24-x)=\sin \lbrace90-9x-2\rbrace \\ 24-x=88-9x \\ 9x-x=88-24 \\ 8x=64 \\ x=8 \end{gathered}

Consider another formula,


\sin (90+\theta)=\cos \theta

This will give the other value of 'x',


\begin{gathered} (1)/(\cos(9x+2))=(1)/(\sin(24-x)) \\ \sin (24-x)=\cos (9x+2) \\ \sin (24-x)=\sin \mleft\lbrace90+(9x+2)\mright\rbrace \\ 24-x=90+9x+2 \\ 9x+x=24-88 \\ 10x=-64 \\ x=-6.4 \end{gathered}

Thus, the given trigonometric equation has two solutions 8 and -6.4

Solve for the first angle as,


\begin{gathered} 9x+2=9(8)+2=72+2=74 \\ 9x+2=9(-6.4)+2=-57.6+2=-55.6 \end{gathered}

Solve for the other angle as,


\begin{gathered} -x+24=-(8)+24=16 \\ -x+24=-(6.4)+24=17.6 \end{gathered}

User Emad Khalil
by
8.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories