Double Angle Calculator
Enter an angle to calculate sin(2x), cos(2x), and tan(2x) using double angle formulas.
Detailed Results
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Request a ToolHow to Use the Double Angle Calculator
This calculator applies the double angle formulas to find sin(2x), cos(2x), and tan(2x) for any angle.
- Enter an angle (x). Type the angle value in the input field.
- Choose degrees or radians. Select the unit that matches your input.
- Read the results. The calculator shows the original trig values (sin x, cos x, tan x) and the double angle results (sin 2x, cos 2x, tan 2x), along with the formulas used.
About Double Angle Formulas
The double angle formulas express trig functions of 2x in terms of trig functions of x. The three primary formulas are: sin(2x) = 2*sin(x)*cos(x), cos(2x) = cos^2(x) - sin^2(x), and tan(2x) = 2*tan(x) / (1 - tan^2(x)). The cosine double angle formula has two equivalent forms: cos(2x) = 2*cos^2(x) - 1 = 1 - 2*sin^2(x). These identities are foundational in trigonometry and are used extensively in calculus, physics, and engineering for simplifying expressions and solving equations.
Frequently Asked Questions
What are the double angle formulas?
The double angle formulas are: sin(2x) = 2*sin(x)*cos(x), cos(2x) = cos^2(x) - sin^2(x), and tan(2x) = 2*tan(x) / (1 - tan^2(x)). These express trig functions of twice an angle in terms of the original angle.
When is tan(2x) undefined?
Tan(2x) is undefined when 1 - tan^2(x) = 0, which means tan(x) = 1 or tan(x) = -1. This occurs at x = 45 degrees and x = 135 degrees (and their coterminal angles). It is also undefined when cos(2x) = 0, which happens at the same angles.
What are the alternative forms of cos(2x)?
The cosine double angle formula has three equivalent forms: cos(2x) = cos^2(x) - sin^2(x) = 2*cos^2(x) - 1 = 1 - 2*sin^2(x). You can derive each from the others using the identity sin^2(x) + cos^2(x) = 1. Each form is useful in different contexts.