tan30°=√3/3; tan45°=1; tan60°=√3; tan90° does not exist.

sin30°=0.5; sin45°=√2/2; sin60°=√3/2; sin90°=1;

cos30°=√3/2; cos45°=√2/2; cos60°=0.5; cos90°=0;

Trigonometric values of some other special angles are shown in the following table:

Trigonometric function is one of the basic elementary functions, is the angle (most commonly used in mathematics radian system, the same below) as the independent variable, the angle corresponds to any angle final side and the unit circle intersection coordinates or the ratio of the function as the dependent variable. It can also be equivalently defined in terms of the lengths of the various line segments associated with the unit circle.

Trigonometric functions play an important role in the study of the properties of geometric shapes such as triangles and circles, and are a fundamental mathematical tool in the study of periodic phenomena. In mathematical analysis, trigonometric functions are also defined as solutions of infinite series or specific differential equations, allowing their values to extend to any real or even complex value.

Extension:

Trigonometry mnemonic:

Trigonometry is a function of a function, quadrant symbols coordinates note. Function image unit circle, period parity increase or decrease now.

The same angle relationship is very important, simplification and proof are needed. Hexagonal vertex, from top to bottom chord cut;

Center of the number one, link vertex triangle. Downward trigonometric sum of squares, inverse relationship is diagonal,

The vertex of any function, equal to the next two roots divided. Induction formula is good, negative to positive and then large to small,

To become an acute angle is good to look up the table, the simplification of the proof of less. Half of two integer multiples, odd to even unchanged,

The latter will be regarded as an acute angle, the sign of the original function judgment. The cosine of the sum of two angles, into a single angle is good to find the value,

Cosine product minus the sine product, the angle of transformation of the many formulas. The sum and difference of the product must be the same name, the reciprocal angle change the name.

Calculate the proof of the angle first, pay attention to the structure of the function name, keep the basic amount of the same, difficult to easy to change.

The inverse principle as a guide, ascending power descending and difference product. The proof of the conditional equation, the idea of the equation shows the way.

The universal formula is not ordinary, into rational formulas first.

The formula is used smoothly and inversely, the use of deformation and clever use;

A plus cosine think cosine, a minus cosine think sine, the power to rise once the angle is halved, the power to rise and fall it for the model;

Trigonometric inverse function, the essence of the angle is to find the trigonometric function, the first trigonometric function, and then determine the range of the angle value;

Using the right triangle, the image of intuitive and good for a change of name, the equation of the triangle, the simple triangles, to the most simplified solution set.

Definition domain and value domain:

sin (x), cos (x) is defined by R, the value domain is [-1,1].

tan(x) has a domain of definition of x not equal to π/2+kπ (k∈Z) and a domain of values of R.

cot(x) has a domain of definition of x not equal to kπ (k∈Z) and a domain of values of R.

y=a-sin(x)+b-cos(x)+c has a domain of values of [ c-√(a² +b& sup2;) , c+√(a² +b²)] with period T=2π/ω.

Trigonometric inverses:

Trigonometric inverses, are multi-valued functions. They are arcsin x, inverse cosine arcsin x, inverse tangent arctan x, inverse cotangent arccot x, etc., each denoting the angle whose sine, cosine, tangent, cotangent, secant, cosecant, and cotangent are x.

In order to limit the inverse trigonometric function to a single-valued function, the value y of the inverse chord function is limited to y = -π/2 ≤ y ≤ π/2, and y is the main value of the inverse chord function, which is recorded as y = arcsin x; accordingly, the main value of the inverse cosine function y = arccos x is limited to 0 ≤ y ≤ π; the main value of the inverse tangent function y = arctan x is limited to -π/2<y<. π/2; the inverse cotangent function y=arccot x has a principal value limit at 0<y<π.

The inverse trigonometric function cannot actually be called a function because it does not satisfy the requirement that an independent variable corresponds to a function value, and its image is symmetric with its original function with respect to the function y=x. The concept was first introduced by Euler, and the form arc + function name was first used to denote the inverse trigonometric function, rather than f-1(x).

There are three main inverse trigonometric functions:

y=arcsin (x), domain of definition [-1,1], domain of values [-π/2,π/2], graph with red lines;

y=arccos (x), domain of definition [-1,1], domain of values [0,π], graph with blue lines;

y=arctan (x ), definition domain (-∞,+∞), value domain (-π/2,π/2), graph with green lines;

sinarcsin(x)=x,definition domain [-1,1], value domain [-π/2,π/2]

Proof method is as follows: set arcsin(x)=y, then sin(y)=x ,substituting these two equations into the above equation can be obtained.

A few others can be obtained in a similar way.

Reference:

Baidu Encyclopedia - Trigonometric Functions