Given that \theta is an acute angle satisfying \sec \theta + \tan \theta = 3, what is the … — JAMB UTME

Given that $\theta$ is an acute angle satisfying $\sec \theta + \tan \theta = 3$ , what is the exact numerical value of $10 \sin \theta$ ?

Answer: 8

The numerical value of

10 \sin \theta

is 8.

Using the standard fundamental identity

\sec^2 \theta - \tan^2 \theta = 1

, we factor it into

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

. Substituting the given value

\sec \theta + \tan \theta = 3

gives

\sec \theta - \tan \theta = \frac{1}{3}

. Solving these linear equations yields

\sec \theta = \frac{5}{3}

(so

\cos \theta = \frac{3}{5}

) and

\tan \theta = \frac{4}{3}

. Using

\sin \theta = \tan \theta \cdot \cos \theta

, we find

\sin \theta = \frac{4}{5} = 0.8

. Multiplying by 10 gives the exact result 8.

Apply the trigonometric identity

\sec^2 \theta - \tan^2 \theta = 1

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

Difference of squares factorization links

\sec \theta + \tan \theta

and

\sec \theta - \tan \theta

as reciprocals.

Substitute

\sec \theta + \tan \theta = 3

to find

\sec \theta - \tan \theta

\sec \theta - \tan \theta = \frac{1}{3}

Dividing both sides of

3(\sec \theta - \tan \theta) = 1

by 3.

Solve the system of equations for

\sec \theta

and

\tan \theta

\sec \theta = \frac{5}{3} \text{ and } \tan \theta = \frac{4}{3}

Adding equations gives

2\sec \theta = \frac{10}{3}

; subtracting gives

2\tan \theta = \frac{8}{3}

Calculate

\sin \theta

and evaluate

10 \sin \theta

\sin \theta = \frac{4}{5} \implies 10 \sin \theta = 8

Since

\cos \theta = \frac{3}{5}

and

\tan \theta = \frac{4}{3}

\sin \theta = \tan \theta \cdot \cos \theta = \frac{4}{5}

Trigonometric identities relating secant and tangent ratios

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