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Solve the following equation on the interval [0°, 360º). Round answers to the nearest tenth. If there is no solution, indicate "No Solution."2sec^2(x) - 13tan(x) = -13

Solve The Following Equation On The Interval 0 360º Round Answers To The Nearest Tenth If There Is No Solution Indicate No Solution2sec2x 13tanx 13 class=

Sagot :

Given

[tex]2\sec ^2(x)-13\tan (x)=-13[/tex]

Add 13 to both sides

[tex]\begin{gathered} 2\sec ^2(x)-13\tan (x)+13=-13+13 \\ 2\sec ^2(x)-13\tan (x)+13=0 \end{gathered}[/tex]

We have that

[tex]\sec ^2(x)=1+\tan ^2(x)[/tex]

So, substitute in the above equation

[tex]2(1+\tan ^2(x))-13\tan (x)+13=0[/tex]

Simplify

[tex]\begin{gathered} 2+2\tan ^2(x)-13\tan (x)+13=0 \\ 15+2\tan ^2(x)-13\tan (x)=0 \end{gathered}[/tex]

Reordering the equation

[tex]2\tan ^2(x)-13\tan (x)+15=0[/tex]

We get a quadratic equation, then solve by factoring

[tex](2\tan (x)-3)(\tan (x)-5)=0[/tex]

Separate the solutions

[tex]\begin{gathered} 2\tan (x)-3=0 \\ 2\tan (x)-3+3=0+3 \\ 2\tan (x)=3 \\ \frac{2\tan (x)}{2}=\frac{3}{2} \\ \tan (x)=\frac{3}{2} \end{gathered}[/tex]

And

[tex]\begin{gathered} \tan (x)-5=0 \\ \tan (x)-5+5=0+5 \\ \tan (x)=5 \end{gathered}[/tex]

Next, solve for x for each solution

[tex]\begin{gathered} \tan (x)=\frac{3}{2} \\ x=\tan ^{-1}(\frac{3}{2}) \\ x=56.3 \end{gathered}[/tex]

And

[tex]\begin{gathered} \tan (x)=5 \\ x=\tan ^{-1}(5) \\ x=78.7 \end{gathered}[/tex]

Answer:

x = 56.3° and x = 78.7°