Given that \theta is an acute angle satisfying the relationship \sec \theta + \tan \theta … — JAMB UTME

Given that $\theta$ is an acute angle satisfying the relationship $\sec \theta + \tan \theta = 3$ , what is the exact value of $5\sin \theta$ ?

Answer: 4

The exact value of

5\sin \theta

is 4.

Using the identity

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

, we deduce

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

. Given

\sec \theta + \tan \theta = 3

, it follows that

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

. Solving the system of equations yields

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

and

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

, which gives

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

. Multiplying by 5 gives the final answer of 4.

Apply the trigonometric Pythagorean identity

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

This relates secant and tangent functions directly.

Factorize the identity and solve for

\sec \theta - \tan \theta

(\sec \theta - \tan \theta)(3) = 1 \implies \sec \theta - \tan \tan \theta = \frac{1}{3}

Using the algebraic identity

a^2 - b^2 = (a - b)(a + b)

Set up a linear system to solve for

\sec \theta

and

\tan \theta

\sec \theta = \frac{5}{3}, \quad \tan \theta = \frac{4}{3}

Adding and subtracting the equations

\sec \theta + \tan \theta = 3

and \sec \theta - \tan \theta = \frac{1}{3} gives the individual function values.

Calculate

\sin \theta

and evaluate

5\sin \theta

\sin \theta = \frac{\tan \theta}{\sec \theta} = \frac{4/3}{5/3} = \frac{4}{5} \implies 5\sin \theta = 4

The quotient of tangent and secant gives sine.

Pythagorean Trigonometric Identities

Draw a right-angled triangle where hypotenuse over adjacent plus opposite over adjacent equals 3:

\frac{c + a}{b} = 3

. By Pythagorean theorem

c^2 - a^2 = b^2

, so

\frac{c - a}{b} = \frac{1}{3}

. Solving yields

a/c = 4/5

, hence

\sin \theta = 4/5

and

5\sin \theta = 4

Estimated Time:2m 0s

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