Sine and cosine

The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and cosine functions. The reciprocal of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as: cos⁡(α) appear to depend on the choice of a right triangle containing an angle of measure sin⁡45∘=cos⁡45∘=22. The following table shows the special value of each input for both sine and cosine with the domain between sin(x) 16π 12 0.5 50g 22 ≈ 0.707 60° 16 12π a, α, sin⁡αa=sin⁡βb=sin⁡γc. This is equivalent to the equality of the first three expressions below: cos⁡(γ)=0, the resulting equation becomes the Pythagorean theorem.

The cross product and dot product are operations on two vectors in Euclidean vector space. The sine and cosine functions can be defined in terms of the cross product and dot product. If θ is the angle between sin⁡(θ)=|𝐚×𝐛||𝐚||𝐛|,cos⁡(θ)=𝐚⋅𝐛|𝐚||𝐛|.

The sine and cosine functions may also be defined in a more general way by using unit circle, a circle of radius one centered at the origin θ with the positive half of the x </math>-"}},"i":0}}]}'>x- and y </math>-"}},"i":0}}]}'>sin⁡(θ), respectively; that is, θ=π2, the point is rotated counterclockwise and stopped exactly on the y </math>-"}},"i":0}}]}'>f is said to be odd if π2. This phase shift can be expressed as xn+1 = cos(xn) with initial value 0</sub>&nbsp;=&nbsp;−1"}},"i":0}}]}'/>x0 = −1 converges to the Dottie number.

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is ddxsin⁡(x)=cos⁡(x),ddxcos⁡(x)=−sin⁡(x). Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. These derivatives can be applied to the first derivative test, according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than equal to zero. It can also be applied to second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero. The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign (0∘<x<90∘ + Decreasing Concave 2nd quadrant, II + Decreasing Concave π<x<3π2 270∘<x<360∘ + Increasing Concave

Both sine and cosine functions can be defined by using differential equations. The pair of y′(θ)=x(θ) and x(0)=1. One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions. It can be interpreted as a phase space trajectory of the system of differential equations y(0)=0 and C denotes the constant of integration. These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the arc length of the sine curve between ∫0t1+cos2(x)dx=2E⁡(t,12), where L=42π3Γ(1/4)2+Γ(1/4)22π=2πϖ+2ϖ≈7.6404 where arcsin(x) and cos⁡:ℝ→ℝ (as well as those functions with the same function rule and domain whose codomain is a subset of sin⁡(0)=0, but also cos⁡:[0,π]→[−1,1]. These are bijective and have inverses: sin−1 for arcsine and cos⁡∘arccos⁡(x)=cos⁡(arccos⁡(x))=xx∈[−1,1],arccos⁡∘cos⁡(θ)=arccos⁡(cos⁡(θ))=θθ∈[0,π].

An acute angle k,

According to Pythagorean theorem, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the Pythagorean trigonometric identity, the sum of a squared sine and a squared cosine equals 1: