Understanding Trigonometry: Finding Sin θ Given Tan θ -√3 and θ > 0

When tackling trigonometric problems, it's essential to understand the underlying principles and apply them step by step. For example, given that tan θ -√3 and θ > 0, we can determine the value of sin θ. Trigonometry, as a branch of mathematics, deals primarily with angles and the relationships involving lengths and properties of triangles. This article will walk you through the process using these guidelines, step by step.

Step-by-Step Solution

Determine the Quadrant

Firstly, we need to identify the quadrant in which θ lies. Since tan θ is negative, we can deduce that θ could be in either the second or fourth quadrant. Given that θ > 0, we need to consider the clockwise direction for the angle, placing θ in the fourth quadrant. This is because the first quadrant has positive values for both sine and cosine, while the fourth quadrant has a positive cosine and a negative sine.

Use the Definition of Tangent

The tangent function is defined as tan θ sin θ / cos θ. Given that tan θ -√3, we can express:

sin θ / cos θ  -√3

From this, we can derive:

sin θ  -√3 * cos θ

Use the Pythagorean Identity

The Pythagorean identity states that sin2θ cos2θ 1. Substituting the expression for sin θ from above, we get:

-√3 * cos θ2   cos θ2  1

The negative sign in front of the √3 complicates the equation, but the cosine squared terms can be combined, leading to:

4 * cos θ2  1

So, we solve for cos θ by taking the square root of both sides:

cos θ  1/2 (since θ in the fourth quadrant, cos θ is positive)

Find sin θ

Finally, substituting the value of cos θ back into the equation for sin θ, we find:

sin θ  -√3 * (1/2)  -√3 / 2

Therefore, the value of sin θ is:

sin θ  -√3 / 2

Additional Considerations

It's important to note that the solution isn't necessarily unique. If tan θ -√3, we can prove that:

cos θ ±1/2 and sin θ ±√3/2.

The selection of signs depends on the specific range of θ. If θ ∈ [-π, π], the value of sin θ is negative in the fourth quadrant, as we have determined. However, in the second quadrant (where θ > π), the value of sin θ would be positive. Thus, the solution is context-dependent.

Using a Calculator

For practical calculations, a scientific calculator can be used. To find sin θ given tan θ -√3 and θ > 0, follow these steps using your calculator:

Enter 3 and follow it with the square root button to get √3. Press the change sign button to get -√3. Ensure your calculator is in degree mode. Press the inverse tangent button (tan^-1) to find the angle -60 degrees. Since your calculator doesn't compute sine of negative angles, use sin(-60) -sin(60) to get sin 60 √3 / 2.

Thus, the value of sin θ is -√3 / 2.

Conclusion

This article provided a comprehensive guide to solving the problem of finding sin θ given tan θ -√3 and θ > 0. Understanding the principles of trigonometry and applying the right trigonometric identities and definitions is crucial for solving such problems accurately. Whether using a calculator or manual methods, always consider the quadrant and context to determine the correct value.