sin30 = 0.5sin 45 =√2/2； sin 60 =√3/2； sin 90 = 1；

cos 30 =√3/2； cos 45 =√2/2； cos60 = 0.5cos 90 = 0；

The trigonometric function values of other special angles are shown in the following table:

Trigonometric function is one of the basic elementary functions, which takes the angle (the most commonly used radian system in mathematics, the same below) as the independent variable, and the angle corresponds to the coordinates where the terminal edge of any angle intersects with the unit circle or its ratio as the dependent variable. It can also be equivalently defined as the lengths of various line segments related to the unit circle.

Trigonometric function plays an important role in studying the properties of geometric shapes such as triangles and circles, and is also a basic mathematical tool for studying periodic phenomena. In mathematical analysis, trigonometric function is also defined as the solution of infinite series or specific differential equation, which allows its value to be extended to any real value or even complex value.

Extended data:

Trigonometric function memory formula:

Trigonometric functions are functions, and quadrant symbols are labeled. Function image unit circle, periodic parity increase and decrease.

The same angle relation is very important, and both simplification and proof are needed. At the vertex of the regular hexagon, cut the chord from top to bottom;

Write the number 1 in the middle to connect the vertex triangles. The sum of the squares of the downward triangle, the reciprocal relationship is diagonal,

Any function of a vertex is equal to the division of the last two. The inductive formula is good, negative is positive and then big and small,

It is easy to look up the table by turning it into an acute angle, and it is essential to simplify the proof. Half of the integer multiple of two, odd complementary pairs remain unchanged,

The latter is regarded as an acute angle, and the sign is judged as the original function. The cosine of the sum of two angles is converted into a single angle, which is convenient for evaluation.

Cosine product minus sine product, angular deformation formula. Sum and difference products must have the same name, and the complementary angle must be renamed.

The calculation proves that the angle is the first, pay attention to the name of the structural function, the basic quantity remains unchanged, and it changes from complexity to simplicity.

Guided by the principle of reverse order, the product of rising power and falling power and difference. The proof of conditional equality, the idea of equation points out the direction.

Universal formula is unusual, rational formula is ahead. The formula is used in the right and wrong direction, and the deformation is used skillfully;

Add cosine to think of cosine, subtract cosine to think of sine, raise the power by half, and raise the power by one norm;

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

Using right triangle, the image is intuitive and easy to rename. The equation of a simple triangle is reduced to the simplest solution set.

Domain and Value Domain:

The domain of sin(x) and cos(x) is r, and the range of values is [- 1, 1].

The definition domain of tan(x) is that x is not equal to π/2+kπ(k∈Z), and the value domain is r.

The definition domain of cot(x) is that X is not equal to kπ(k∈Z), and the value domain is r.

The range of y = a sin (x)+b cos (x)+c is [c-√ (a&; sup2+b & amp； sup2)，c+√(a & amp； sup2+b & amp； Sup2)] Period T=2π/ω.

Inverse function of trigonometric function:

The inverse function of trigonometric function is a multivalued function. They are arcsine x, anti-cosine anti-cosine x, anti-tangent anti-tangent x, anti-tangent anti-tangent x and so on. , which respectively represent sine angle, cosine angle, tangent angle, cotangent angle, secant angle and cotangent angle.

In order to define the inverse trigonometric function as a single-valued function, the value y of the arcsine function is defined as y=-π/2≤y≤π/2, and y is the principal value of the arcsine function, denoted as y = arcsinx. Accordingly, the principal value of the inverse cosine function y=arccos x is limited to 0 ≤ y ≤π; The principal value of arctangent function y=arctan x is limited to -π/2.

In fact, the inverse trigonometric function can't be called a function, because it doesn't meet the requirement that the independent variable corresponds to the function value. Its image and its original function are symmetrical about the function y = X, and its concept was first put forward by Euler. The inverse trigonometric function was first expressed in the form of arc+function name, instead of f- 1(x).

There are three main inverse trigonometric functions:

Y=arcsin(x), domain [- 1, 1], range [-π/2, π/2], and red line is used for images;

Y=arccos(x), domain [- 1, 1], range [0, π], blue line for image;

Y=arctan(x), domain (-∞, +∞), range (-π/2, π/2), and the image is represented by green lines;

Sinarcsin(x)=x, domain [- 1, 1], range [-π/2, π/2]

The proof method is as follows: let arcsin(x)=y, then sin(y)=x, and substitute these two formulas into the above formula.

Several others can be obtained in a similar way.

References:

Baidu encyclopedia-trigonometric function