Cos 2 Theta Calculator
Enter an angle to compute cos(2theta) using the cosine double-angle identity. Switch between degrees, radians, and pi-radians, or pick an alternative formula mode to work from sin(theta) or cos(theta) directly. All three equivalent forms of the identity are shown side by side so you can verify the result and follow every step of the math.
Formula
Worked example
For theta = 30 deg: sin(30 deg) = 0.5, cos(30 deg) = sqrt(3)/2 ~ 0.8660. Form 1: 0.8660^2 - 0.5^2 = 0.7500 - 0.2500 = 0.5000. Form 2: 2 * 0.8660^2 - 1 = 1.5000 - 1 = 0.5000. Form 3: 1 - 2 * 0.5^2 = 1 - 0.5 = 0.5000. All three give cos(60 deg) = 0.5.
What is the cos(2theta) identity?
The cosine double-angle identity expresses cos(2theta) in terms of the trigonometric functions of theta. It comes directly from the cosine addition formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B) when you set A = B = theta: cos(2theta) = cos(theta + theta) = cos(theta)*cos(theta) - sin(theta)*sin(theta) = cos^2(theta) - sin^2(theta). Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, you can substitute to get two alternative forms: replace sin^2(theta) with 1 - cos^2(theta) to get cos(2theta) = 2cos^2(theta) - 1, or replace cos^2(theta) with 1 - sin^2(theta) to get cos(2theta) = 1 - 2sin^2(theta). All three forms are equivalent and will give the same numerical answer for any angle theta.
When to use each form
The three forms of the identity are interchangeable in principle, but each is more convenient in specific situations. Use cos^2(theta) - sin^2(theta) when you already know both sin(theta) and cos(theta), for example when solving a right triangle. Use 2cos^2(theta) - 1 when you want to rewrite an expression involving only cosines and remove the squared term, or when only cos(theta) is known and you prefer not to compute sin(theta). Use 1 - 2sin^2(theta) when you want to eliminate the cosine entirely and work only with sines, which is useful in integration and in rewriting power-reduction expressions. In physics, the power-reduction forms are used to simplify the energy stored in oscillating systems and to convert between instantaneous and average power in AC circuits.
Power-reduction and half-angle connections
The sine-only form of the double-angle identity, cos(2theta) = 1 - 2sin^2(theta), can be rearranged to the power-reduction formula sin^2(theta) = (1 - cos(2theta)) / 2. Similarly, the cosine-only form gives cos^2(theta) = (1 + cos(2theta)) / 2. These power-reduction forms are essential in calculus when integrating even powers of sine and cosine, because the squared functions cannot be integrated directly but their double-angle equivalents can. They also lead directly to the half-angle identities: replacing theta with theta/2 in the power-reduction formulas gives sin(theta/2) = sqrt((1 - cos(theta)) / 2) and cos(theta/2) = sqrt((1 + cos(theta)) / 2).
Applications in science and engineering
The double-angle cosine identity appears across mathematics and science. In classical mechanics, the angle dependence of projectile range involves cos(2theta) because the horizontal range R = (v^2 * sin(2theta)) / g, and the condition for maximum range (theta = 45 deg, cos(2theta) = 0) follows from differentiating. In AC electrical engineering, the instantaneous power in a resistive circuit is p(t) = V_m^2 * cos^2(omega*t) / R, which the power-reduction form rewrites as a constant term plus a term oscillating at double frequency 2*omega. In optics, the Malus law for polarized light can be expressed in double-angle form. In structural analysis, the Mohr-circle transformation for stress states uses both the sin(2theta) and cos(2theta) forms to rotate stresses between coordinate systems.
cos(2theta) for standard angles
| theta | theta (degrees) | 2theta (degrees) | cos(2theta) exact | cos(2theta) decimal |
|---|---|---|---|---|
| 0 | 0 deg | 0 deg | 1 | 1.000000 |
| pi/12 | 15 deg | 30 deg | sqrt(3)/2 | 0.866025 |
| pi/8 | 22.5 deg | 45 deg | sqrt(2)/2 | 0.707107 |
| pi/6 | 30 deg | 60 deg | 1/2 | 0.500000 |
| pi/4 | 45 deg | 90 deg | 0 | 0.000000 |
| pi/3 | 60 deg | 120 deg | -1/2 | -0.500000 |
| 3pi/8 | 67.5 deg | 135 deg | -sqrt(2)/2 | -0.707107 |
| 5pi/12 | 75 deg | 150 deg | -sqrt(3)/2 | -0.866025 |
| pi/2 | 90 deg | 180 deg | -1 | -1.000000 |
| 2pi/3 | 120 deg | 240 deg | -1/2 | -0.500000 |
| 3pi/4 | 135 deg | 270 deg | 0 | 0.000000 |
| 5pi/6 | 150 deg | 300 deg | 1/2 | 0.500000 |
| pi | 180 deg | 360 deg | 1 | 1.000000 |
Exact and decimal values of cos(2theta) at commonly used angles. These appear frequently in algebra, geometry, and physics problems.
Frequently asked questions
Why are there three different formulas for cos(2theta)?
They are all the same identity written in different ways. Starting from cos(2theta) = cos^2(theta) - sin^2(theta), you apply the Pythagorean identity sin^2 + cos^2 = 1 twice: substituting sin^2 = 1 - cos^2 gives the cosine-only form 2cos^2 - 1, and substituting cos^2 = 1 - sin^2 gives the sine-only form 1 - 2sin^2. The three forms are mathematically identical and will always produce the same number for any valid angle.
How do I calculate cos(4theta) using this identity?
Apply the identity twice. First compute cos(2theta) from theta. Then treat 2theta as your new angle and apply the identity again: cos(4theta) = cos(2*(2theta)) = 2cos^2(2theta) - 1. For example, cos(4*30 deg) = cos(120 deg). From the first step, cos(2*30 deg) = cos(60 deg) = 0.5. Then 2*(0.5)^2 - 1 = 0.5 - 1 = -0.5. That matches cos(120 deg) = -0.5.
What if I only know sin(theta)? Can I still compute cos(2theta)?
Yes. Use the sine-only form: cos(2theta) = 1 - 2sin^2(theta). You just square sin(theta), multiply by 2, and subtract from 1. For example, if sin(theta) = 0.6, then cos(2theta) = 1 - 2*(0.6)^2 = 1 - 0.72 = 0.28. Note this gives the unique value of cos(2theta) regardless of which quadrant theta is in, because squaring removes the sign.
What is the range of cos(2theta)?
cos(2theta) is a cosine function, so its range is always between -1 and 1 inclusive, just like cos(theta). The value -1 occurs when 2theta is an odd multiple of 180 deg, meaning theta is an odd multiple of 90 deg. The value +1 occurs when 2theta is a multiple of 360 deg, meaning theta is a multiple of 180 deg. The value 0 occurs when 2theta is an odd multiple of 90 deg, meaning theta is an odd multiple of 45 deg.
How is this different from cos(theta/2)?
The double-angle identity gives cos(2theta) from cos(theta) and sin(theta). The half-angle identity goes the opposite direction: cos(theta/2) = sqrt((1 + cos(theta)) / 2), giving you a function of half the angle from the full angle. This calculator computes the double-angle result; to find a half-angle cosine, divide your input angle by 2 before entering it, or use a dedicated half-angle tool.
Does cos(2theta) equal 2*cos(theta)?
No, this is a common mistake. The cosine function is not linear, so cos(2theta) is not the same as 2*cos(theta). For example, cos(2*60 deg) = cos(120 deg) = -0.5, but 2*cos(60 deg) = 2*0.5 = 1.0. The correct relationship is the double-angle identity cos(2theta) = cos^2(theta) - sin^2(theta).