Solving Trigonometric Equations Using Given Values of Sine

Understanding the relationship between sine and cosine, and how to apply trigonometric identities, is key to solving complex trigonometric equations. This article will walk you through a detailed example where we are given the value of sine (sin) and asked to find the values of sin2 and cos2, as well as cos(2φ) and sin(2φ).

Given: sin(φ) 5/13, where sin(φ) > 0

Since sin(φ) 5/13 and it is positive, is either in the first or second quadrant.

Step 1: Find cos(φ)

Using the Pythagorean identity, we can find the value of cos(φ).

sin2(φ) cos2(φ) 1

cos(φ) sqrt{1 - sin2(φ)} sqrt{1 - (5/13)2} sqrt{1 - 25/169} sqrt{144/169} ±12/13

Given that sin(φ) > 0, we must consider the possible quadrants for φ. In the first quadrant, cos(φ) is positive, so:

cos(φ) 12/13

Step 2: Calculate sin2(φ)

The square of sine is:

sin2(φ) (5/13)2 25/169

Step 3: Calculate cos2(φ)

The square of cosine is:

cos2(φ) (12/13)2 144/169

Step 4: Use Double-Angle Identities

The double-angle identity for cosine is:

cos(2φ) cos2(φ) - sin2(φ)

Substituting the values, we get:

cos(2φ) 144/169 - 25/169 119/169

Step 5: Find sin(2φ)

The double-angle identity for sine is:

sin(2φ) 2sin(φ)cos(φ)

Substituting the values, we get:

sin(2φ) 2 × (5/13) × (12/13) 120/169

In summary:

sin(φ) 5/13 cos(φ) 12/13 sin2(φ) 25/169 cos2(φ) 144/169 cos(2φ) 119/169 sin(2φ) 120/169

These calculations demonstrate the application of trigonometric identities and the Pythagorean theorem to solve for unknown values in trigonometric equations. Understanding these identities and their applications is crucial for advanced studies in mathematics and physics.