Sin2x Identity

Sin2x is a popular trigonometry identity. Trigonometry, as it suggests, is all about triangles, mostly right-angled triangles. It is a system that helps us to work out missing or unknown side lengths or angles in a triangle.

Sine, Cosine & Tangent are some of the functions used for finding the sides of a triangle with their given universal values.

History

The Greek mathematician Hipparchus produced the first known table of chords in about 140 BC. Although these tables have not survived, it is claimed that Hipparchus wrote twelve books of tables of chords, making Hipparchus the founder of trigonometry.

Now let us see about Sin2x identity.

Sin2x Formula 

Sin2x is one of the crucial identities in trigonometry that can be expressed in different ways. 

Sin2x identity can be expressed in terms of different trigonometric functions such as sine & cosine and tangent.

\(\sin(2x) = 2 \sin(x) \cos(x)\)

\(\sin(2x) = \frac{2 \tan(x)}{1 + \tan^2x}\)

Derivation of Sin2x Identity

\(\Large{\sin(2x) = \frac{2 \tan(x)}{1 + \tan^2x}}\)

\(
\begin{align}
\sin(2x) & = 2 \sin(x) \cos(x) \\
& = \frac{2 \sin(x) \cos^2(x)}{\cos(x)} \\
& = (\frac{2 \sin(x)}{\cos(x)})(sec^2(x)) \\
& = \frac{2 \tan(x)}{1 + \tan^2(x)} \\
\end{align}
\)

Sin2x Identity Questions

Question 1. Determine the value of \(\sin(2x)\) if \(\cos(x) = \frac{3}{4}\).

Solution. From identity  \(\sin^2(x) + \cos^2(x) = 1\)

\(\sin^2(x) = 1 – \cos^2(x)\)

\(\sin^2(x) = 1 – \frac{9}{16}\)

\(\sin^2(x) = \frac{7}{16}\)

\(\sin(x) = \frac{\sqrt{7}}{4}\)

\(\sin(2x)     = 2 \sin(x) \cos(x)\)

\(\sin(2x) = 2 \frac{\sqrt{7}}{4} \frac{3}{4}\)

\(\sin(2x) = \frac{3\sqrt{7}}{8}\)

Question 2. If \(\tan(t) = \frac{4}{3}\) find the value of \(\sin(2t)\).

Solution.

\(\sin(2x) = \frac{2 \tan(x)}{1 + \tan^2(x)}\)

\(\sin(2x) = \frac{2 \cdot \frac{4}{3}}{1 + (\frac{4}{3})^2}\)

\(\sin(2x) = \frac{24}{25}\)

FAQs

Related Articles

1. Cos2x Identity
2. Tan2x Identity