Finding sin 2θ Given tan θ 5/12 and θ in Quadrant III

Given that tan θ 5/12 and θ lies in the third quadrant, we aim to find the value of sin 2θ. This problem involves the use of trigonometric identities and understanding the signs of trigonometric functions in different quadrants.

Using Trigonometric Identities

To find the value of sin 2θ, we will start by determining the values of sin θ and cos θ using the given information and trigonometric identities.

Step 1: Determine sin θ and cos θ

Given that tan θ 5/12, we can construct a right triangle where the opposite side is 5 and the adjacent side is 12. Using the Pythagorean theorem, we can find the hypotenuse:

[ hypotenuse sqrt{5^2 12^2} sqrt{25 144} sqrt{169} 13 ]

Step 2: Determine the Signs of sin θ and cos θ

Since θ is in the third quadrant, both sin θ and cos θ are negative.

[ sin θ -frac{5}{13} ]

[ cos θ -frac{12}{13} ]

Step 3: Use the Double-Angle Identity for Sine

The double-angle identity for sine is:

[ sin 2θ 2 sin θ cos θ ]

Substituting the values of sin θ and cos θ:

[ sin 2θ 2 left(-frac{5}{13}right) left(-frac{12}{13}right) 2 times frac{60}{169} frac{120}{169} ]

Alternative Method Using tan θ

We can also use the double-angle identity involving tangent:

[ sin 2θ frac{2 tan θ}{1 tan^2 θ} ]

Given tan θ 5/12:

[ sin 2θ frac{2 left(frac{5}{12}right)}{1 left(frac{5}{12}right)^2} frac{frac{10}{12}}{1 frac{25}{144}} frac{frac{10}{12}}{frac{169}{144}} frac{10}{12} times frac{144}{169} frac{10 times 12}{169} frac{120}{169} ]

Conclusion

Thus, the value of sin 2θ given that tan θ 5/12 and θ lies in the third quadrant is:

[ sin 2θ frac{120}{169} ]

Understanding Trigonometric Signatures in Each Quadrant

It's important to remember the signs of trigonometric functions in each quadrant:

First Quadrant: All functions are positive. Second Quadrant: sin and csc are positive. Third Quadrant: tan and cot are positive, while sin and csc are negative. Fourth Quadrant: cos and sec are positive.

Since θ is in the third quadrant, tan and cot are positive, while sin and cos are negative.

Key Takeaways

1. Using Pythagorean theorem to find the hypotenuse of the reference triangle. 2. Applying the signs of trigonometric functions in the third quadrant. 3. Utilizing trigonometric identities to find complex trigonometric values.

For more information on trigonometry, visit our comprehensive guides on:

Trigonometric Identities Quadrants and Signs of Trigonometric Functions Solving Trigonometric Equations