Understanding the Trigonometric Problem

The problem asks us to find the value of the expression $(3 – \tan A) (2 + \cos A)$, given that $\sin A = \frac{4}{5}$. To solve this, we first need to find the values of $\cos A$ and $\tan A$ using the given value of $\sin A$. We will assume that angle A is in a quadrant where sine, cosine, and tangent are defined and positive for simplicity in calculating the primary values, typically the first quadrant.

Finding Cos A from Sin A

We can use the fundamental trigonometric identity relating sine and cosine:

$\sin^2 A + \cos^2 A = 1$

Substitute the given value $\sin A = \frac{4}{5}$ into the identity:

$(\frac{4}{5})^2 + \cos^2 A = 1$

$\frac{16}{25} + \cos^2 A = 1$

Now, isolate $\cos^2 A$ by subtracting $\frac{16}{25}$ from 1:

$\cos^2 A = 1 - \frac{16}{25}$

To subtract the fractions, find a common denominator:

$\cos^2 A = \frac{25}{25} - \frac{16}{25}$

$\cos^2 A = \frac{25 - 16}{25}$

$\cos^2 A = \frac{9}{25}$

Take the square root of both sides to find $\cos A$. Assuming A is in the first quadrant, $\cos A$ is positive:

$\cos A = \sqrt{\frac{9}{25}}$

$\cos A = \frac{3}{5}$

Calculating Tan A using Sin A and Cos A

The tangent of an angle A is defined as the ratio of the sine of A to the cosine of A:

$\tan A = \frac{\sin A}{\cos A}$

Substitute the values we have found: $\sin A = \frac{4}{5}$ and $\cos A = \frac{3}{5}$:

$\tan A = \frac{\frac{4}{5}}{\frac{3}{5}}$

To simplify this compound fraction, we can multiply the numerator by the reciprocal of the denominator:

$\tan A = \frac{4}{5} \times \frac{5}{3}$

Cancel out the common factor 5 in the numerator and the denominator:

$\tan A = \frac{4}{3}$

Evaluating the Given Trigonometric Expression

Now we need to evaluate the expression $(3 – \tan A) (2 + \cos A)$. We have determined that $\tan A = \frac{4}{3}$ and $\cos A = \frac{3}{5}$. Substitute these values into the expression:

Expression $= (3 – \frac{4}{3}) (2 + \frac{3}{5})$

First, evaluate the term inside the first bracket:

$3 – \frac{4}{3}$

Convert 3 to a fraction with a denominator of 3:

$3 = \frac{3 \times 3}{3} = \frac{9}{3}$

So, $3 – \frac{4}{3} = \frac{9}{3} – \frac{4}{3} = \frac{9 - 4}{3} = \frac{5}{3}$

Next, evaluate the term inside the second bracket:

$2 + \frac{3}{5}$

Convert 2 to a fraction with a denominator of 5:

$2 = \frac{2 \times 5}{5} = \frac{10}{5}$

So, $2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$

Now, multiply the results of the two brackets:

Expression $= (\frac{5}{3}) \times (\frac{13}{5})$

Multiply the numerators and the denominators:

Expression $= \frac{5 \times 13}{3 \times 5}$

Cancel the common factor 5 from the numerator and the denominator:

Expression $= \frac{13}{3}$

The value of the expression $(3 – \tan A) (2 + \cos A)$ is $\frac{13}{3}$.

Revision Table: Basic Trigonometric Ratios

Additional Information on Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that are true for every value of the variable for which the functions are defined. The identity $\sin^2 A + \cos^2 A = 1$ is one of the most important Pythagorean identities. It is derived directly from the Pythagorean theorem applied to the coordinates on a unit circle or the sides of a right-angled triangle. Understanding these identities allows us to simplify complex trigonometric expressions and solve various problems in trigonometry, calculus, and physics. For instance, knowing one trigonometric ratio of an angle often allows you to find the other ratios using these identities, as demonstrated in this problem where we found $\cos A$ and $\tan A$ from $\sin A$. Other key identities relate the reciprocal functions (cosecant, secant, cotangent) to the basic functions (sine, cosine, tangent).