If \sin \theta = \frac{3}{5} for an acute angle \theta, what is the exact value of 5 \cos … — JAMB UTME

If $\sin \theta = \frac{3}{5}$ for an acute angle $\theta$ , what is the exact value of $5 \cos \theta + 4 \tan \theta$ ?

Answer: 7

The exact value of

5 \cos \theta + 4 \tan \theta

is 7.

For an acute angle

\theta

with

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

, the corresponding right triangle has opposite side = 3, hypotenuse = 5, and adjacent side = 4. Using trig definitions,

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

and

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

. Evaluating

5 \cos \theta + 4 \tan \theta

gives

5\left(\frac{4}{5}\right) + 4\left(\frac{3}{4}\right) = 4 + 3 = 7

Find the adjacent side of the right-angled triangle.

Adjacent side =

\sqrt{5^2 - 3^2} = 4

By the Pythagorean theorem (

a^2 + b^2 = c^2

), where opposite = 3 and hypotenuse = 5.

Determine the values of

\cos \theta

and

\tan \theta

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

and

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

Using fundamental trigonometric definitions: cosine is adjacent/hypotenuse and tangent is opposite/adjacent.

Substitute these values into

5 \cos \theta + 4 \tan \theta

and simplify.

5\left(\frac{4}{5}\right) + 4\left(\frac{3}{4}\right) = 4 + 3 = 7

Performing simple multiplication and addition gives 7.

Basic Trigonometric Ratios in Right Triangles

Question