Understanding Cos 2 Theta
Cos 2 theta, written cos(2θ), is the cosine of a doubled angle. It comes from the cosine addition formula cos(A + B) = cosA·cosB − sinA·sinB. Setting A = B = θ gives cos(2θ) = cosθ·cosθ − sinθ·sinθ = cos²θ − sin²θ. Combined with the Pythagorean identity sin²θ + cos²θ = 1, this identity can be rewritten in two more useful forms.
The double-angle formula (three equivalent forms)
- cos(2θ) = cos²θ − sin²θ — the direct definition, useful when both sinθ and cosθ are known.
- cos(2θ) = 2cos²θ − 1 — obtained by substituting sin²θ = 1 − cos²θ; useful when only cosθ is known.
- cos(2θ) = 1 − 2sin²θ — obtained by substituting cos²θ = 1 − sin²θ; useful when only sinθ is known.
A fourth form uses the tangent: cos(2θ) = (1 − tan²θ) / (1 + tan²θ). This calculator evaluates cos(2θ) directly from θ by doubling the angle and applying the cosine function, then reports sin(2θ) = 2 sinθ cosθ and tan(2θ) = sin(2θ) / cos(2θ) alongside it.
Degrees vs. radians
Choose the unit that matches your problem before reading the result. Conversion: radians = degrees × π/180, and a full circle is 2π radians = 360°. The calculator converts your angle to radians internally regardless of which unit you pick, so the trigonometric functions are always evaluated correctly.
Common applications
- Power-reduction and half-angle formulas in calculus, which are derived directly from cos(2θ) = 1 − 2sin²θ and cos(2θ) = 2cos²θ − 1
- Simplifying trigonometric integrals and Fourier series terms that involve cos²θ or sin²θ
- Physics and engineering problems with doubled phase angles, such as wave interference, AC power calculations, and rotational motion
Frequently Asked Questions
What is the formula for cos 2 theta?
How do I find cos 2 theta if I know tan theta?
Does this calculator use degrees or radians?
Why is cos(2θ) sometimes negative even when cos(θ) is positive?
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |