Table of contents
1 Measures of angles: degrees and radians
1.2 Arc Length and Chord Length
1.3.1 The principle of arc length and radian definition
1.4 Conversion of angle and radian
1.5 Conversion of angles and radians in EXCEL
2 Calculation of trigonometric functions
3 Advanced trigonometric functions and formulas
1 Measures of angles: degrees and radians
There are usually two units of measurement for angles, one is the angle system, and the other is the radian system.
1.1 angle angle
1.1.1 Definition
- What is an angle: A measure of the amount by which any one of two intersecting straight lines is superimposed on the other, and the rotation is carried out on the plane of the two straight lines and around the point of intersection.
- On the Mesopotamian plains, the ancient Babylonians in BC called the circumference 1 degree (recorded as 1°), and there are two units of "minute" and "second" under the degree, 60 minutes 1 60 seconds is 1 minute.
- The unit of angle is degree, which is the unit used to measure the size of an angle, and the symbol is °.
- A circle angle is divided into 360 equal parts, and each part is defined as 1 degree (1°). The number 360 is used for the circumference angle because it is easily divisible.
- In addition to 1 and itself, 360 has 22 true factors, including numbers from 2 to 10 other than 7, so the angles of many special angles are integers.
1.1.2 Formula
- Angle is a mathematical concept.
- The size of an angle can be described, that is, the amount by which either of two intersecting lines must be turned to coincide with the other.
- The angle calculation formula is tanB=(x2-x1)/(y2-y1)
1.1.2 Angle value range
- Theoretically, there is no value range, 180, 360, 720 degrees
- But according to the specific graphics, the angle of some graphics has a range.
- If it is a circle, then the angle ∈ [0,360]
1.2 Arc Length and Chord Length
- The arc length is the length of the arc, which is the curve.
- The chord length is the line segment connecting the end points of two line segments in graphics such as a sector, which is a straight line.
- In a circle, chord length = circumference length = 2Πr
- Length, of course, there is no upper limit to the length, [0, +∞]
1.3 radians rad
1.3.1 The principle of arc length and radian definition
- Why is there a radian
- Angle is a measure of 360 degrees, and radius is measured by length, which is a completely different measure. Calculation is more troublesome
- The idea of establishing arc is to unify arc length and radius. Both of these two units are measured by length, and there is only one standard of length.
- In this way, radians = arc length / radius, which is also equal to a unit of length
1.3.2 Definition
- In mathematics and physics, a radian is a unit of measure for an angle. The abbreviation is rad.
- Definition: An arc whose arc length is equal to its radius subtends a central angle of 1 radian. That is to say, two rays shoot out from the center of the circle to the circumference, forming an arc with an included angle and an arc opposite to the included angle. When the length of this arc is exactly equal to the radius of the circle, the angle between the two rays is 1).
- So, a radian is the value of the arc length divided by the radius in a circle,
1.3.3 Range of values
- |radian|=arc length/radius
1.3.4 Range of values
- Theoretically, radians have no range, and correspond to lengths. There is no upper limit for length, and radians can also have no upper limit.
- However, in some graphics, the value of radians has a limited range
- The arc length of the circle [0,2Πr] corresponds to the arc of the circle [0,2Π]
1.4 Conversion of angle and radian
- convert angle to radian
- Because a perfect circle, 360 degrees = 2Π radians
Formula deformation derivation
- 360 degrees = 2Π radians = 2Π * (180/Π) degrees = 360 degrees
- 2Π弧度 = 2Π * (180/Π) 度
- 1 radian = (180/Π) degrees
- Therefore, its conversion formula
- Π=3.1415926, EXCEL returns Π with pi()
- radians=angle*PI()/180 = angle*0.017453293
- Angle=radian*180/PI() = radian*57.29577951
- So 1 radian = 180/ Π = 57.29 degrees
- So 1 degree = Π/180 = 0.017 radians
1.5 Conversion of angles and radians in EXCEL
- Conversion formula
- radians = angle*PI()/180
- Radian = RADIANS (angle) EXCEL's built-in function
- The specific calculation table below
2 Calculation of trigonometric functions
2.1 Trigonometric functions
2.1.1 Definition
- angle
- arc length
- sin()
- cos()
- tan()
- cot()
- sec()
- css()
2.1.2 Range of values
- The range of trigonometric functions is [-1,1]
2.2 To calculate trigonometric functions in EXCEL, you need to use radian values, such as sin (radian)
- To calculate trigonometric functions in EXCEL, you need to use radian values instead of angles
- Such as sin (radian)
- The range of trigonometric functions is [-1,1]
3 Advanced trigonometric functions and formulas
sum of two angles formula
- sin(A+B) = sinAcosB+cosAsinB
- sin(A-B) = sinAcosB-cosAsinB
- cos(A+B) = cosAcosB-sinAsinB
- cos(AB) = cosAcosB+sinAsinB
- tan(A+B) = (tanA+tanB)/(1-tanB)
- tan(AB) = (tanA-tanB)/(1+tanB)
- cot(A+B) = (cotAcotB-1)/(cotB+cotA)
- cot(AB) = (cotAcotB+1)/(cotB-cotA)
double angle formula
- tan2A = 2tanA/(1-tan² A)
- Sin2A = 2SinA CosA
- Cos2A = Cos^2 A–Sin² A
- =2Cos² A—1
- =1—2sin^2 A
sum and difference product
- sin(a)+sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]
- sin(a)-sin(b) = 2cos[(a+b)/2]sin[(a-b)/2]
- cos(a)+cos(b) = 2cos[(a+b)/2]cos[(a-b)/2]
- cos(a)-cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]
- tanA+tanB=sin(A+B)/cosAcosB
accumulation and difference
- sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)]
- cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)]
- sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)]
- cos(a)sin(b) = 1/2*[sin(a+b)-sin(a-b)]