Trigonometric identity transformation formula A formula for transforming a * Sin & + b * cos & into basic form The derivation process of Union

Trigonometric identity transformation formula A formula for transforming a * Sin & + b * cos & into basic form The derivation process of Union

A * Sin & + b * cos & = sqrt (a ^ 2 + B ^ 2) sin (& + θ) [where Tan (θ) = B / a]
For supplementary questions: if you want to deduce the process, you need to offer a reward of 100 points

Trigonometric identity transforms all formulas

Trigonometric function of sum and difference of two angles: cos (α + β) = cos α · cos β - sin α · sin β cos (α - β) = cos α · cos β + sin α · sin β sin (α + β) = sin α · cos β + cos α · sin β sin (α - β) = sin α · cos β - cos α · sin β Tan (α + β) = (Tan α + Tan β) / (1-ta

What are the universal formulas of mathematics, not those of trigonometric functions,

Multiplication and factorization A2-B2 = (a + b) (a-b) A3 + B3 = (a + b) (a2-ab + B2) a3-b3 = (a-b) (A2 + AB + B2)
Trigonometric inequality | a + B | ≤| a | + | B | A-B | ≤| a | + | B | a | ≤ B-B ≤ a ≤ B
|a-b|≥|a|-|b| -|a|≤a≤|a|
Solutions of quadratic equation of one variable - B + √ (b2-4ac) / 2A - B-B + √ (b2-4ac) / 2A
The relation between root and coefficient X1 + x2 = - B / a X1 * x2 = C / A
Discriminant b2-4a = 0 note: the equation has two equal real roots
B2-4ac > 0 note: the equation has a real root
b2-4ac

It's about triangles
In the triangle ABC, the opposite sides of angles a, B and C are a, B and C respectively, and satisfy (2c-a) CoSb bcosa = 0
(1) If B = 7, a + C = 13, find the area of triangle ABC
(2) Find the value range of sina + sin (c-sextant) with root sign 3 times

According to the cosine theorem, CoSb = (a ^ 2 + C ^ 2-B ^ 2) / 2Ac, cosa = (b ^ 2 + C ^ 2-A ^ 2) / 2BC, a ^ 2 + C ^ 2-B ^ 2 = AC
(1) If we bring in B = 7, we get a ^ 2 + C ^ 2-ac = 49. That is, (a + C) ^ 2-3ac = 49, bring in a + C = 13, we get AC = 40, and have a + C = 13, so a and C are 5 and 8 respectively. According to cosine theorem, CoSb = (a ^ 2 + C ^ 2-B ^ 2) / 2Ac = (25 + 64-49) / (2 * 5 * 8) = 1 / 2, SINB = 2 / 2 root sign 3, triangle area is 1 / 2 * acsinb = 1 / 2 * 5 * 8 * 2 root sign 3 = 10 times root sign 3
(2) If a ^ 2 + C ^ 2-B ^ 2 = AC, CoSb = (a ^ 2 + C ^ 2-B ^ 2) / 2Ac = 1 / 2, B = one-third faction. A + C = two-thirds faction. Replace a = two-thirds faction-c with a = two-thirds faction-c. the root sign 3 times sin (two-thirds faction-a) + sin (c-sixths faction) is expanded to get the root sign 3 * sinc + COSC, and the angle formula 2 times sin (c + three thirds faction). The range of C is (0, two-thirds faction), The value range of sin (c + Third) is (1 / 2,1], so the value range of the original formula is (1,2]

A bag of rice weighs 40 kg. If you eat it in 20 days, how many parts of this bag of rice do you eat every day? How many kg do you eat every day?

Take the whole bag of rice as the unit "1" with 1 △ 20 = one twentieth, and the average daily consumption is 40 △ 20 = 2 (kg)

A bag of rice weighs 100 kg. How many parts of a bag of rice do you eat? How many parts are left?

Two fifths, three fifths

1. A bag of rice is eaten in 25 days. How many parts of the bag of rice are eaten every day and how many parts of the bag of rice are eaten in 5 days

1. A bag of rice is finished in 25 days, and the average daily consumption is 1 / 25 of the bag of rice, and 1 / 5 of the bag of rice in 5 days

A bag of 20 kg rice, the first time to eat half of all, the second time to eat the remaining half, how much of this bag of rice is left?

What's left
=1-2 / 1 - (1-2 / 1) × 1 / 2
=1-2 / 1-4 / 1
=1 / 4

A bag of rice ate 20% in the first day, 30 kg in the second day, 1 / 2 of this bag of rice in two days, how many kg in total

30÷（1/2-20%)=100
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If a bag of rice is 50 kg and it is planned to be eaten in 15 days, then it will be eaten in 10 days______ You can eat 45% of this bag of rice in one day

If a bag of rice is 50kg, we plan to eat it in 15 days, and eat 1015 of this bag of rice in 10 days, that is 23; another 45 = 810, actually 110 of this bag of rice every day, and 45 of this bag of rice in 8 days