Cosine Function cos(x): Graph, Derivative, Integral and Properties

The cosine function takes an angle x, measured in radians, and returns the x-coordinate of the point at that angle on the unit circle. Its graph is the same wave as sine, shifted left by a quarter period, so it starts at its peak instead of at zero.

Definition of cos x

Travel a distance x counterclockwise around the unit circle from (1, 0). The point you reach is (\cos x, \sin x), so

For an acute angle in a right triangle this is the ratio \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}.

Graph of y = cos x

Key values of cosine

Notice the row is the sine row read backward on \left[0, \frac{\pi}{2}\right], because \cos x = \sin\left(\frac{\pi}{2} - x\right).

Properties of the cosine function

• Domain and range. All real numbers in, values in [-1, 1] out.
• Period. \cos(x + 2\pi) = \cos x, and A\cos(bx) has amplitude |A| and period \frac{2\pi}{|b|}.
• Zeros. \cos x = 0 exactly when x = \frac{\pi}{2} + k\pi for an integer k. These are the places where tangent and secant have vertical asymptotes.
• Even symmetry. \cos(-x) = \cos x.
• Identities. \sin^2 x + \cos^2 x = 1, and the double angle formula \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x.

Derivative of cos x

The minus sign makes sense on the graph: just after x = 0 cosine is falling, and -\sin x is negative there. At x = \pi cosine bottoms out, and -\sin\pi = 0. The second derivative is -\cos x, so cosine, like sine, satisfies y'' = -y.

With the chain rule, \frac{d}{dx}\cos(u) = -\sin(u)\cdot u'.

Integral of cos x

For example, \int_0^{\pi/2}\cos x\,dx = \sin\frac{\pi}{2} - \sin 0 = 1.

Worked examples

Example 1: chain rule. Differentiate f(x) = \cos(4x - 1). The inside derivative is 4, so f'(x) = -4\sin(4x - 1).

Example 2: power of cosine. Differentiate g(x) = \cos^3 x = (\cos x)^3. The outer power gives 3\cos^2 x and the inside gives -\sin x:

Example 3: u-substitution. Evaluate \int x\cos(x^2)\,dx. Let u = x^2, so du = 2x\,dx:

Example 4: a tangent line. At x = \frac{\pi}{2}, \cos x = 0 and the slope is -\sin\frac{\pi}{2} = -1. The tangent line is y = -\left(x - \frac{\pi}{2}\right).

Common mistakes

1. Dropping the minus sign. \frac{d}{dx}\cos x = -\sin x, not \sin x. A useful memory aid: derivatives of the "co" functions (cosine, cotangent, cosecant) all carry a minus sign.
2. Adding a minus sign to the integral. \int\cos x\,dx = \sin x + C with no minus sign.
3. Misreading \cos^2 x. \cos^2 x means (\cos x)^2, not \cos(\cos x) and not \cos(x^2).
4. Forgetting the inside derivative. \frac{d}{dx}\cos(2x) = -2\sin(2x), not -\sin(2x).

How cosine connects to other topics

Cosine is the denominator of tangent and the reciprocal of secant (see reciprocal trig functions). Restricted to [0, \pi], it has the inverse arccosine. In Calculus 2 you will integrate powers of cosine using the identity \cos^2 x = \frac{1 + \cos 2x}{2}, and meet its Taylor series \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots. For more practice differentiating it, see the guide to derivatives of trig functions.