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When you dive into trigonometry or calculus, you’ll encounter functions like sine, cosine, and tangent. Guessing the value of a trigonometric equation with a chart or a calculator can be tedious or even impossible. That’s why trigonometric identities – short, proven relationships – are essential for simplifying and solving these equations.
Double‑angle identities let you express sin(2θ), cos(2θ), and tan(2θ) in terms of single‑angle functions. They’re a subset of the more general sum and difference formulas.
Two equivalent forms exist:
\\(\\sin(2\\theta)=2\\sin(\\theta)\\cos(\\theta)\\)
\\(\\sin(2\\theta)=\\frac{2\\tan(\\theta)}{1+\\tan^2(\\theta)}\\)
Cosine can be written in several useful ways:
\\(\\cos(2\\theta)=\\cos^2(\\theta)-\\sin^2(\\theta)\\)
\\(\\cos(2\\theta)=2\\cos^2(\\theta)-1\\)
\\(\\cos(2\\theta)=1-2\\sin^2(\\theta)\\)
\\(\\cos(2\\theta)=\\frac{1-\\tan^2(\\theta)}{1+\\tan^2(\\theta)}\\)
Only one practical form is used:
\\(\\tan(2\\theta)=\\frac{2\\tan(\\theta)}{1-\\tan^2(\\theta)}\\)
These identities are invaluable when you need to rewrite a trigonometric expression so that only one type of function remains. The angle symbol can be any letter – θ, α, x, or β – because the identity holds for all angles.
Rewrite cos 2x + sin 2x using only sin x and cos x:
\\(\\cos(2x)+\\sin(2x)=\\bigl(2\\cos^2(x)-1\\bigr)+\\bigl(2\\sin(x)\\cos(x)\\bigr)\\)
\\(\\quad=2\\cos(x)\\bigl(\\cos(x)+\\sin(x)\\bigr)-1\\)
1. Simplify 2 cos² 32 – 1:
\\(2\\cos^2(32)-1=\\cos(2\\times32)=\\cos(64)\\)
2. Simplify 2 sin α cos α where α = β⁄2:
\\(2\\sin(α)\\cos(α)=\\sin(2\\alpha)=\\sin(\\beta)\\)
