Solving for sin 2θ when tan θ a: A Step-by-Step Guide

This article will guide you through the process of solving for sin 2θ when tan θ a, using fundamental trigonometric identities and the double angle formula. Understanding these principles can be invaluable for students and professionals working with trigonometry, particularly in fields related to engineering, physics, and mathematics.

Introduction to the Problem

In trigonometry, the relationship between trigonometric functions (like sine, cosine, and tangent) is often expressed in terms of angles. When given the tangent of an angle (tan θ a), we can find other trigonometric values, such as sin 2θ, using algebraic manipulation and the Pythagorean identity.

Step-by-Step Solution

We are given that tan θ a. Let's break down the solution into manageable steps to understand how to find sin 2θ.

Step 1: Express sin θ and cos θ in Terms of a

Starting with the definition of tangent:

tan θ {sin θ}{cos θ} a

This implies:

sin θ a cos θ

Step 2: Use the Pythagorean Identity

The Pythagorean identity states:

sin2 θ cos2 θ 1

Substituting sin θ a cos θ into this identity:

(a cos θ)2 cos2 θ 1

This can be simplified to:

a2 cos2 θ cos2 θ 1

Factoring out cos2 θ:

cos2 θ (a2 1) 1

Solving for cos2 θ:

cos2 θ 1}{a2 1}

Finally, taking the square root to find cos θ:

cos θ ±1}{√(a2 1)}

Here, the ± sign indicates that both the positive and negative values are possible, depending on the quadrant of θ.

Step 3: Use the Sine Identity in Term of Cosine

Using the identity sin θ a cos θ:

sin θ a (1}{√(a2 1)}) a}{√(a2 1)}

Step 4: Substitute into the Double Angle Formula

The double angle formula for sine is:

sin 2θ 2 sin θ cos θ

Substituting our values:

sin 2θ 2 (a}{√(a2 1)}) (1}{√(a2 1)})

This simplifies to:

sin 2θ 2a}{a2 1}

Conclusion

We have successfully determined that sin 2θ 2a}{a2 1} when tan θ a. This result demonstrates the power of trigonometric identities in simplifying complex expressions.