Understanding the Problem

The problem asks us to find the value of the expression \(\tan x + \sin x\), given that \(\sec x = \frac{25}{24}\) and the angle $x$ lies in the fourth quadrant. To solve this, we need to find the individual values of \(\tan x\) and \(\sin x\) first, using the given information.

Step 1: Finding cos x from sec x

We know that \(\sec x\) is the reciprocal of \(\cos x\). The relationship is given by the identity \(\cos x = \frac{1}{\sec x}\).

Given \(\sec x = \frac{25}{24}\), we can find \(\cos x\):

\(\cos x = \frac{1}{\frac{25}{24}} = \frac{24}{25}\)

Since $x$ is in the fourth quadrant, the cosine value is positive, which aligns with our calculated value of \(\frac{24}{25}\).

Step 2: Finding sin x using Trigonometric Identity

We can use the fundamental trigonometric identity \(\sin^2 x + \cos^2 x = 1\) to find \(\sin x\).

Substitute the value of \(\cos x\) we just found:

\(\sin^2 x + \left(\frac{24}{25}\right)^2 = 1\)

\(\sin^2 x + \frac{576}{625} = 1\)

Now, solve for \(\sin^2 x\):

\(\sin^2 x = 1 - \frac{576}{625} = \frac{625 - 576}{625} = \frac{49}{625}\)

Taking the square root of both sides gives us \(\sin x\):

\(\sin x = \pm\sqrt{\frac{49}{625}} = \pm\frac{7}{25}\)

We are given that $x$ lies in the fourth quadrant. In the fourth quadrant, the sine function is negative. Therefore, we must choose the negative value for \(\sin x\):

\(\sin x = -\frac{7}{25}\)

Step 3: Finding tan x using sin x and cos x

The tangent of an angle is defined as the ratio of its sine to its cosine: \(\tan x = \frac{\sin x}{\cos x}\).

Using the values of \(\sin x\) and \(\cos x\) we found:

\(\tan x = \frac{-\frac{7}{25}}{\frac{24}{25}}\)

Multiplying the numerator by the reciprocal of the denominator:

\(\tan x = -\frac{7}{25} \times \frac{25}{24} = -\frac{7}{24}\)

In the fourth quadrant, the tangent function is negative, which is consistent with our result of \(-\frac{7}{24}\).

Step 4: Calculating tan x + sin x

Now we have the values for \(\tan x\) and \(\sin x\). We can find their sum:

\(\tan x + \sin x = \left(-\frac{7}{24}\right) + \left(-\frac{7}{25}\right)\)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 24 and 25 is \(24 \times 25 = 600\).

Convert the fractions to have a denominator of 600:

\(-\frac{7}{24} = -\frac{7 \times 25}{24 \times 25} = -\frac{175}{600}\)

\(-\frac{7}{25} = -\frac{7 \times 24}{25 \times 24} = -\frac{168}{600}\)

Now, add the converted fractions:

\(\tan x + \sin x = -\frac{175}{600} - \frac{168}{600} = \frac{-175 - 168}{600} = \frac{-343}{600}\)

So, the value of \(\tan x + \sin x\) is \(-\frac{343}{600}\).

Revision Table: Trigonometric Signs in Quadrants

It's important to remember which trigonometric functions are positive in each quadrant. A common mnemonic is "CAST" (starting from Quadrant IV and moving counterclockwise): Cosine (and secant) is positive in IV, All are positive in I, Sine (and cosecant) is positive in II, Tangent (and cotangent) is positive in III.

Additional Information: Reciprocal Identities

Besides the Pythagorean identities like \(\sin^2 x + \cos^2 x = 1\), remember the reciprocal identities which are useful for finding other trigonometric values when one is given.

• \(\sec x = \frac{1}{\cos x}\)
• \(\csc x = \frac{1}{\sin x}\)
• \(\cot x = \frac{1}{\tan x}\)

Also, the quotient identities:

• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\cot x = \frac{\cos x}{\sin x}\)

These identities, along with the knowledge of signs in different quadrants, are fundamental tools for solving trigonometric problems like this one.