Exploring the Relationship Between Trigonometric Functions and Solving for SinX, CosX / 1 - SinX When Given TanX 12/5

In this article, we delve into the relationship between trigonometric functions and how to solve for sinXcosX / 1 - sinX when tanX 12/5.

Understanding the Problem

The problem at hand is to find the value of sinXcosX / 1 - sinX given that tanX 12/5. This requires understanding the properties of the trigonometric functions sine, cosine, and tangent. We start by recalling the basic definitions and identities.

Using Trigonometric Identities

We know that the tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle:

tanX sinX / cosX

Given tanX 12/5, we can express sinX and cosX in terms of this ratio.

Deriving sinX and cosX

Starting from the identity:

tanX 12/5

We can write:

sinX 12k and cosX 5k for some constant k

Using the Pythagorean identity:

sin^2X cos^2X 1

We substitute the values of sinX and cosX to find k and hence the values of sinX and cosX.

Solving for k

We have:

(12k)^2 (5k)^2 1

Expanding and simplifying:

144k^2 25k^2 1
169k^2 1

k^2 1/169

Since k is positive, we have:

k 1/13

Thus, the values are:

sinX 12/13 and cosX 5/13

Evaluating the Expression

Now, we can substitute these values into the expression:

sinX cosX / 1 - sinX (12/13) * (5/13) / (1 - 12/13)
(12/13) * (5/13) / (1/13)
(12/13) * (5/13) * (13/1)

12/13 * 5 60/13

However, the problem can also be approached using a geometric method involving a right triangle.

Geometric Approach with Right Triangles

Given tanX 12/5, we can draw a right-angled triangle where the opposite side is 12 and the adjacent side is 5. By the Pythagorean theorem, we can find the hypotenuse:

hypotenuse √(12^2 5^2) √(144 25) √169 13

Thus, we have:

sinX opposite / hypotenuse 12/13

cosX adjacent / hypotenuse 5/13

Now, we can use these values to evaluate the expression:

sinX cosX / 1 - sinX (12/13) * (5/13) / (1 - 12/13)
(12/13) * (5/13) / (1/13) 60/13

Multiple Solutions Considering Sign

It's important to consider the signs of sine and cosine. There are two possible cases:

Both Sine and Cosine are Negative

If both sinX and cosX are negative, we have:

sinX -12/13

cosX -5/13

Substituting these into the expression:

fx (-12/13) * (-5/13) / (1 - (-12/13))
(60/169) / (25/13) 60/169 * 13/25 17/25

Both Sine and Cosine are Positive

If both sinX and cosX are positive, we have:

sinX 12/13

cosX 5/13