Finding the Exact Value of Tan 2α Given sin α 12/13

In this article, we will explore the process of finding the exact value of Tan 2α given that sin α 12/13 and α is an acute angle. We will use the double angle formula for tangent, along with some trigonometric identities, to arrive at the solution. This process involves the use of right triangles and the Pythagorean theorem.

Step-by-Step Solution

We start with the given information:

sin α 12/13

Our goal is to find Tan 2α. First, we need to find the value of Tan α.

1. Finding Cos α

Using the Pythagorean identity:

sin2α cos2α 1

We can solve for Cos α as follows:

Cos2α 1 - sin2α

Cos2α 1 - (12/13)2

Cos2α 1 - 144/169

Cos2α (169 - 144)/169

Cos2α 25/169

Cos α sqrt(25/169) 5/13

2. Finding Tan α

Now we can find Tan α using the definition of tangent:

Tan α sin α / cos α

Tan α (12/13) / (5/13) 12/5

Tan α 12/5

3. Using the Double Angle Formula for Tangent

Now, we use the double angle formula for tangent:

Tan 2α 2 * Tan α / (1 - Tan2α)

Substituting the value of Tan α into the formula:

Tan 2α 2 * (12/5) / (1 - (12/5)2)

Tan 2α (2 * 12/5) / (1 - 144/25)

Finding a common denominator for the denominator:

1 - 144/25 (25/25) - (144/25) (25 - 144)/25 -119/25

Substituting back into the formula:

Tan 2α (2 * 12/5) / (-119/25)

Multiplying the fractions:

Tan 2α (2 * 12/5) * (-25/119)

Tan 2α (24/5) * (-25/119) 24 * (-5) / 119 -120/119

4. Conclusion

The exact value of Tan 2α is:

Tan 2α -120/119

Visualization with a Right Triangle

We can visualize the situation with a right triangle in the first quadrant, where:

Opposite side 12 Hypotenuse 13 Adjacent side 5 (calculated as the square root of 169 - 144)

Thus, we get the triangle with sides AB 5, BC 12, and CA 13, where angle A is α, and B is the right angle.

Final Calculation

Using the double angle identity for tangent:

Tan 2α [2 * Tan α] / [1 - Tan2α]

Substituting the values:

Tan 2α [2 * 12/5] / [1 - (12/5)2] [24/5] / [25/25 - 144/25] [24/5] / [-119/25]

Calculating the final value:

Tan 2α (24/5) * (-25/119) -120/119

The exact value of Tan 2α is:

Tan 2α -120/119

Therefore, the value of Tan 2α is approximately -1.0084 to four decimal places.

Conclusion

We have successfully used trigonometric identities and the Pythagorean theorem to find the exact value of Tan 2α. This process highlights the importance of understanding both the theoretical and practical applications of trigonometry in solving complex problems.