Proof of the sum angle formula, and the derivation process of the sum angle formula

Method 1: Vector proof.

in a planar Cartesian coordinate system.

, take the x-axis as the starting edge, make the angle , and the angle is recorded as the unit vector of the end edge.

is a, b, then a=(cos, sin), b=(cos, sin).

a·b=|a||b|cos

and a·b = cos ·cos + sin ·sin

And|a|=|b|=1

cos=cos(α-=cosα·cosβ+sinα·sinβ

Replace with - to get cos( +=cos ·cos -sin ·sin

by induction formula.

6，sin(α-=-=-

cos（α+/2）·cosβ+sin（α+/2）·sinβ】

-sinα·cosβ+cosα·sinβ】

sinα·cosβ-cosα·sinβ

In the same way, sin( +=sin ·cos +cos +cos ·sin

and tan( -= sin( - cos( -= (sin ·cos -cos ·sin ) (cos ·cos +sin ·sin )

In addition to cos ·cos, we get tan( -=(tan -tan ) (1+tan ·tan )

In the same way, tan( +=(tan +tan ) (1-tan ·tan )

Certification. Method 2: Geometric proof.

See the figure on the right for details, replacing - with , from the resulting difference angle formula.

cos(α-=cosβcosα+sinβsinα

obtained: cos( +=cos cos -sin sin

om=ob+bm

ob+cp|oa|cosα+|ap|sin (sin here refers to sin cap, according to the triangular relationship.)

cap= aom=) can be rolled out

cosβcosα+sinβsinα

The derivation process of the sum angle formula is as follows:

sin (αsinα·cosβ +cosα·sinβ sin (αsinα·cosβ -cosα·sinβ

cos (αcosα·cosβ -sinα·sinβ cos (αcosα·cosβ +sinα·sinβ

tan (αtanα-tanβ) 1+tanα·tanβ)tan (αtanα+tanβ) 1-tanα·tanβ)

aob = aop = op|=1, unit circle.

cos(α-cos∠pom = om = ob+cp

In AOB, Bishumin ob = oa·cos

In apc cp = ap·sin cap, and cap = cp = ap·sin

om = oa·cos α ap·sinα

In AOP, OA = COS ·OP = COS

In AOP, the hand branch ap = sin ·op = sin

cos(α-cosβ·cos α sinβ·sinα

Replace - with , you get:

cos (αcosα·cosβ -sinα·sinβ

tan( -sin( -cos( -

sinα·cosβ-cosα·sinβ)/cosα·cosβ+sinα·sinβ)

In addition to cos ·cos, we get tan( -=(tan -tan ) (1+tan ·tan )

In the same way, tan( +=(tan +tan ) (1-tan ·tan )

The sum angle formula, also known as the addition theorem of trigonometric functions, is the trigonometric function of the sum (difference) of several angles, and the relationship expressed by the trigonometric function of each angle in it.

The most commonly used formulas are as follows.

The commonly used formula for <> and angle is sina 2 + cosa 2 = 1. The sum and angle formula, also known as the addition theorem of trigonometric functions, is the relationship between the sum (difference) of several angles and the trigonometric function expressed by the trigonometric function of each of the angles. Angular functions are a class of functions in mathematics that belong to the transcendental functions of elementary functions.

Trigonometric functions play an important role in the study of the properties of geometric shapes such as triangles and circles, and are also the basic mathematical tools for studying periodic phenomena in research. In mathematical analysis, trigonometric functions are also defined as solutions to infinite series or specific differential equations, allowing their values to be extended to arbitrary real values, even complex values.

The two straight lines are parallel, the inner misalignment angles are equal, and the isotope angles are equal; Congruent triangles, similar triangles correspond to equal angles; The vertex angles are equal; The outer angles of the triangle are equal to the sum of the two inner angles that are not adjacent; The first relative of the same horn or the complementary angle of the same angle is equal; Co-angles of the same or equal angles are equal; The parallelograms are equal in diagonals.

In geometry, an angle is a geometric object made up of two rays with a common endpoint. These two rays are called the edges of the corner, and their common endpoints are called the vertices of the corner. Ordinary angles are assumed to be on the Euclidean plane, but angles can also be defined in Euclidean geometry.

Angles have a wide range of applications in geometry and trigonometry.

Euclid, the father of geometry, defined an angle as the relative slope of two non-parallel straight lines in a plane. Procruz thought that an angle might be a trait, a quantifiable quantity, or a relation. Eudemo considered an angle to be a deviation from a relative straight line, and Qaboos of Antioch considered an angle to be the space between two intersecting straight lines.

Euclid considered angles to be a relation, but his definition of right, acute, or obtuse angles was quantified.

Summary. Angle BOC = 60°, while OB = OC, so angle OBC = Angle OCB = 60°. So.

Hello, little master, please describe the problem you are experiencing.

Please wait, it may take a while for you to write the answer here.

1) Because the small object is made of a flat bridge and a flat air, then Xinhe H1-H2=1 2GT t=2(H1-H2) G=2(40-20) 10=4s

On the second question, how do you prove that the two corners I marked are equal?

Hmm, please wait.

Little master: May I ask which corner is it called bac **It's a little unclear.

Is it horn BOC and horn obc?

Hello little lord, these two horns you marked do not want to wait.

The angular BOC and the angular OBC are equal, and the other angle you mark is smaller than the angular OBC

Angle BOC = 60°, while OB = OC, so angle OBC = Angle OCB = 60°. So.

So how to do the second question.

Please wait. Little master, I wrote it down with a pen, the process is a bit troublesome, mainly to find the horizontal distance between A and B. I'm sorry to wait a minute.

Let's see if I can understand it.

Here we need to consider that the title says that it enters the arc exactly in the tangent direction of b.

Are you doing something wrong, the answer is not this.

What's your answer?

Sorry, the longitudinal velocity formula is wrong there.

The answer is elastic potential energy ep=800j

Oh oh t=2 I was wrong when I first asked the equation there.

The second question is okay, but the t is not right, I substituted t=4 and the result is wrong.