Deducing the Value of (2theta) Given (sin theta frac{5}{13}) in an Obtuse Angle Context

Introduction:

When dealing with trigonometric functions, particularly in the context of obtuse angles, understanding the relationships between sine, cosine, and doubling an angle becomes essential. This article provides a detailed exploration of how to determine ( sin 2theta ) given that ( sin theta frac{5}{13} ) in an obtuse angle scenario. It also includes practical examples and an explanation of how to verify the result using a TI-83 or TI-84 calculator.

Understanding the Problem

The angle ( theta ) is obtuse, meaning its terminal arm is in the second quadrant of the unit circle. In the second quadrant, the sine of an angle is positive, while the cosine is negative. Given that ( sin theta frac{5}{13} ), we first need to determine the cosine of ( theta ).

Using the Pythagorean identity:

( cos^2 theta 1 - sin^2 theta )

We can substitute the value of ( sin theta ):

( cos^2 theta 1 - left(frac{5}{13}right)^2 1 - frac{25}{169} frac{144}{169} )

Taking the square root of both sides, we get:

( cos theta pm frac{12}{13} )

Since ( theta ) is in the second quadrant, ( cos theta ) is negative, thus:

( cos theta -frac{12}{13} )

Applying the Double-Angle Formula

The double-angle formula for sine is:

( sin 2theta 2 sin theta cos theta )

Substituting the known values of ( sin theta ) and ( cos theta ):

( sin 2theta 2 cdot frac{5}{13} cdot -frac{12}{13} -frac{120}{169} )

Verification Using a Calculator

For verification, we can use a TI-83 or TI-84 calculator. First, calculate ( theta ) using the inverse sine function:

Step 1: Enter ( sin^{-1} left( frac{5}{13} right) ) on the calculator.

Step 2: Convert the angle to radians and use the double-angle formula to find ( sin 2theta ).

Example with (sin theta frac{3}{5})

Consider another scenario where ( sin theta frac{3}{5} ). This angle is part of a 3-4-5 right triangle. Thus, we can determine the cosine and tangent of ( theta ) as follows:

( cos theta frac{4}{5} ) ( tan theta frac{3}{4} )

For ( theta ) as an acute angle, ( sin 2theta ) can be found using:

( sin 2theta 2 sin theta cos theta 2 cdot frac{3}{5} cdot frac{4}{5} frac{24}{25} )

Note that for better precision, use the double-angle formula directly:

( cos^2 theta 1 - sin^2 theta 1 - left( frac{3}{5} right)^2 frac{16}{25} ) ( cos theta frac{4}{5} )

Then calculate: