Question

find an angle \\( \theta \\) that makes the statement true.
\\( \sec \left(3 \theta-76^{\circ}\
ight)=\csc \left(\theta+26^{\circ}\
ight) \\)
choose the correct answer below.
\\( \bigcirc \\) a. \\( 15^{\circ} \\) \\( \bigcirc \\) b. \\( -15^{\circ} \\)
\\( \bigcirc \\) c. \\( 55^{\circ} \\) \\( \bigcirc \\) d. \\( 14^{\circ} \\)
\\( \bigcirc \\) e. \\( 105^{\circ} \\) \\( \bigcirc \\) f. \\( 90^{\circ} \\)
\\( \bigcirc \\) g. \\( 35^{\circ} \\) \\( \bigcirc \\) h. \\( 76^{\circ} \\)

Explanation:

Step1: Use co - function identity

We know that \(\sec x=\csc(90^{\circ}-x)\). So, if \(\sec(3\theta - 76^{\circ})=\csc(\theta + 26^{\circ})\), then \(3\theta-76^{\circ}=90^{\circ}-(\theta + 26^{\circ})\)

Step2: Expand the right - hand side

Expand \(90^{\circ}-(\theta + 26^{\circ})\) to get \(90^{\circ}-\theta - 26^{\circ}=64^{\circ}-\theta\). So the equation becomes \(3\theta-76^{\circ}=64^{\circ}-\theta\)

Step3: Solve for \(\theta\)

Add \(\theta\) to both sides: \(3\theta+\theta-76^{\circ}=64^{\circ}-\theta+\theta\), which simplifies to \(4\theta-76^{\circ}=64^{\circ}\). Then add \(76^{\circ}\) to both sides: \(4\theta-76^{\circ}+76^{\circ}=64^{\circ}+76^{\circ}\), so \(4\theta = 140^{\circ}\). Divide both sides by \(4\): \(\theta=\frac{140^{\circ}}{4}=35^{\circ}\)

Answer:

G. \(35^{\circ}\)