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.
5 + 5 tan² θ = 6 sec θ

Accepted Answer

Start by rewriting the given equation: $$5 + 5 	an$$^{2}$$ 	heta = 6 \sec 	heta. $$Recall the Pythagorean identity relating tangent and secant: $$1 + 	an$$^{2}$$ 	heta = \sec$$^{2}$$ 	heta. $$Use this to express $$	an$$^{2}$$ 	heta in $$terms of $$\sec$$^{2}$$ 	heta as$$ $$	an$$^{2}$$ 	heta = \sec$$^{2}$$ 	heta - 1. $$Substitute $$	an$$^{2}$$ 	heta = \sec$$^{2}$$ 	heta - 1$$ into the original equation to get $$5 + 5(\sec$$^{2}$$ 	heta - 1) = 6 \sec 	heta. $$Simplify the equation to form a quadratic in $$\sec 	heta$$: $$5 + 5 \sec$$^{2}$$ 	heta - 5 = 6 \sec 	heta$$, which reduces to $$5 \sec$$^{2}$$ 	heta = 6 \sec 	heta. $$Rewrite the equation as $$5 \sec$$^{2}$$ 	heta - 6 \sec 	heta = 0$$, factor it, and solve for $$\sec 	heta. $$Then, find the corresponding values of $$	heta by $$considering the definition $$\sec 	heta = \frac{1}{\cos 	heta}$$ and solving for $$	heta in $$the specified domain, ensuring to express solutions in the least possible nonnegative angle measures.

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Key Concepts

Trigonometric Identities

Solving Trigonometric Equations

Angle Measurement and Conversion