Formulas of trigonometric functions includes sum-difference angle formula, sum-difference product formula, product sum-difference formula, angle doubling formula, induction formula and so on. Here are some commonly used formulas of trigonometric functions:

And angle formula: sin(α+β)=sinαcosβ+cosαsinβ, sin(α-β)=sinαcosβ-cosαsinβ, cos(α+β)=cosαcosβ-sinαsinβ, cos(α-β)=cosαcosβ+sinαsinβ.

Sum product formula: sin α+sin β = 2 sin ((α+β)/2) cos ((α-β)/2), sin α-sin β = 2 cos ((α+β)/2) sin ((α-β)/2), cos α+cos β = 2 cos.

Sum and difference formula of products: sin α cos β =1/2 * [sin (α+β)+sin (α-β)], cos α sin β =1/2 * [cos (α+β)-cos (α-β)].

Double angle formula: sin2α=2sinαcosα, cos2α=cos? α-sin？ α=2cos？ α- 1= 1-2sin？ α，tan2α=2tanα/( 1-tan？ α)

Inductive formula: sin(π/2+α)=cosα, cos(π/2+α)=-sinα, tan(π/2+α)=cotα, cot(π/2+α)=tanα.

Trigonometric function is an important branch of mathematics, which includes sine, cosine and tangent functions. These functions are widely used in geometry and algebra.

First, let's look at the definition of trigonometric function. In right triangle, sine function is defined as the ratio of opposite side to hypotenuse, cosine function is defined as the ratio of adjacent side to hypotenuse, and tangent function is defined as the ratio of opposite side to adjacent side. These definitions can be understood intuitively by drawing pictures.

Next, let's look at the properties of trigonometric functions. First of all, we know that the periods of sine and cosine functions are all 2π/360 degrees = 180 degrees/π radians. Secondly, the images of sine and cosine functions are symmetrical about the origin. In addition, the sum and difference formulas of sine and cosine functions are also very important, which can help us simplify the complex trigonometric function expressions.

Besides sine and cosine functions, there is also an important trigonometric function-tangent function. The image of tangent function is a straight line passing through the origin, and its periodicity is π radian. Tangent function also has many applications in physics and engineering, such as calculating the inclination of a slope or the voltage in a circuit.

Finally, let's look at the application of trigonometric function. In geometry, trigonometric functions can be used to solve various trigonometric problems, such as calculating angles or side lengths. In algebra, trigonometric functions can be used to solve equations or take derivatives. In addition, trigonometric functions are also widely used in signal processing, image processing and other fields.