Let tan θ=(5)/(12) with θin QI and find the following. sin2...

Key Concepts

Double Angle Identity

Double angle identities, particularly the double angle formula for sine (sin 2? = 2 sin ? cos ?), allow the transformation of trigonometric expressions involving double angles into products of functions of the original angle. This identity is central to solving for sin 2? in the given problem by expressing it in terms of sin ? and cos ? derived from the triangle.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, are foundational elements in understanding the relationships between the angles and sides of right triangles. In the context of this problem, knowing tan ? helps establish the ratio between the opposite and adjacent sides, which is essential for finding other trigonometric values.

Right Triangle Construction

Representing the given tangent ratio through a right triangle is a key step. By assigning appropriate lengths to the opposite and adjacent sides according to the tangent definition, one can utilize the geometric relationships inherent in a right triangle to determine the hypotenuse and subsequently compute other trigonometric functions.

Pythagorean Theorem

The Pythagorean theorem is used to calculate the hypotenuse of the right triangle once the lengths of the opposite and adjacent sides are known. This theorem, stating that the square of the hypotenuse equals the sum of the squares of the other two sides, is critical for deriving the sine and cosine values needed for subsequent computations.

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