Question

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

3 csc² x/2 = 2 sec x

Accepted Answer

Rewrite the given equation $$3 \csc$$^{2}$$ \frac{x}{2} = 2 \sec x in $$terms of sine and cosine functions. Recall that $$\csc 	heta = \frac{1}{\sin 	heta}$$ and $$\sec 	heta = \frac{1}{\cos 	heta}$$, so the equation becomes $$3 \left(\frac{1}{\sin$$^{2}$$ \frac{x}{2}}ight) = 2 \left(\frac{1}{\cos x}ight). $$Multiply both sides of the equation by $$\sin$$^{2}$$ \frac{x}{2} \cos x to $$clear the denominators, resulting in an equation involving $$\cos x$$ and $$\sin$$^{2}$$ \frac{x}{2}$$ without fractions. Use the double-angle identity for cosine: $$\cos x = 1 - 2 \sin$$^{2}$$ \frac{x}{2}$$, to express $$\cos x in $$terms of $$\sin$$^{2}$$ \frac{x}{2}. $$Substitute this into the equation to have a single trigonometric function variable. Let $$t = \sin$$^{2}$$ \frac{x}{2}$$ and rewrite the equation as a polynomial in $$t. $$Solve this polynomial equation for $$t to $$find possible values of $$\sin$$^{2}$$ \frac{x}{2}. $$For each valid solution of $$t$$, find $$\sin \frac{x}{2}$$ and then solve for $$x by $$considering the general solutions for sine. Remember to express $$x in $$radians within the least possible nonnegative angle measures, and also convert to degrees if required, rounding as specified.

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Key Concepts

Trigonometric Identities

Solving Trigonometric Equations

Angle Measures and Conversion