1, the main theorems for solving oblique triangles: sine theorem and cosine theorem, cosine projection formula and various types of area formulas.
2. Four kinds of problems that can be solved: (1) Knowing two angles and one edge (2) Knowing two edges and included angles (3) Knowing three edges (4) Knowing the diagonal of two edges and one of them.
(2) Solving right triangle
1, the main theorem of solving right triangle: in right triangle ABC, the right angle is angle C, angle A and angle B are its two acute angles, and the sum of angles A, B, C, (1) and angle B is 90 degrees; (2) Pythagorean theorem: the square of A plus the square of +b = the square of C; (3) The sine of angle A is equal to ratio C, the cosine of angle A is equal to ratio C, the sine of angle B is equal to ratio C, and the cosine of angle B is equal to ratio C; (4) formula of area s=ab/2; In addition, there are projective theorems, inner and outer radii of inscribed circles.
2. Solve four kinds of right-angled triangles: (1) Two right-angled sides are known: according to Pythagorean theorem, first find the hypotenuse, use trigonometric function to find one of the two acute angles, then use complementary relation to find the other angle or use trigonometric function to find two of the two acute angles; (2) Given a right-angled edge and a hypotenuse, find another right-angled edge according to Pythagorean theorem, and the problem is transformed into (1); (3) When a right-angled edge and an acute angle are known, we can find another acute angle, the hypotenuse with sine or cosine, and the other right-angled edge with Pythagorean theorem; (4) When the hypotenuse and an acute angle are known, the opposite side of the known angle is calculated first, and then the other right-angled side is calculated according to Pythagorean theorem, and the problem is transformed into (1).
(1) Two sine theorems for solving triangle problems;
1. Given two angles and an arbitrary edge, find the other two edges and an angle.
2. Know the diagonal of two angles and one of them, and find the other angles.
(2) Two kinds of cosine theorems are used to solve the triangle problem:
1. Find a triangle from the three known sides.
2. Knowing the angles between two sides and them, find the third side and the other two angles.
1. In a certain measure, if A is 40 southeast of B, then B is A's ().
A. 40 degrees north by west B. 50 degrees north by east
C. 50 degrees northwest. 50 southwest.
A: A.
2. It is known that the distance between A and B is 10 km, and the distance between B and C is 20 km. Now ∠ ABC = 120 is measured, so the distance between a and c is ().
10 km
C. 105 km D. 107 km
Analysis: choose D. According to cosine theorem:
AC2=AB2+BC2-2AB？ BCcos∠ABC。
∫AB = 10，BC=20，∠ ABC = 120，
∴ac2= 102+202-2× 10×20×cos 120 = 700。
∴AC= 107.
3. On a 20-meter-high observation platform, the elevation angle of the top of the opposite water tower is 60, and the depression angle of the tower bottom is 45. The bottom of the observation platform and the tower bottom are on the same ground plane, so the height of the water tower is _ _ _ _ _ _ _ m. 。
Analysis: h = 20+20 tan 60 = 20 (1+3) m.
Answer: 20( 1+3)
As shown in the figure, a ship is sailing eastward at the speed of 15 km per hour. At location a, the ship saw lighthouse B 60, 60 east of north. After driving for 4 hours, the ship arrived at location C and saw the east-by-north 15 lighthouse. Find the distance between the ship and the lighthouse at this time.
Solution: BCsin∠BAC=ACsin∠ABC,
And < BAC = 30, AC=60,
∠ABC= 180 -30 - 105 =45。
∴BC=302.
That is, the distance between the ship and the lighthouse is 302 kilometers.
Essential knowledge points in mathematics circle
1. cycle
In a plane, a closed curve formed by a moving point rotating around a certain point with a certain length is called a circle. A circle has countless axes of symmetry.
2. Related characteristics of the circle
(1) diameter
The line segment connecting the center of the circle and any point on the circle is called radius, and the letter is expressed as R.
The line segment passing through the center of the circle with both ends on the circle is called the diameter, and the letter is denoted as d.
A straight line with a diameter is the symmetry axis of a circle. In the same circle, the diameter of the circle is d=2r.
(2) Chords
A line segment connecting any two points on a circle is called a chord. The longest chord in the same circle is the diameter. The straight line with the diameter is the symmetry axis of the circle, so there are countless symmetry axes of the circle.
(3) Arc shape
The part between any two points on a circle is called an arc, which is represented by "⌒".
An arc larger than a semicircle is called an optimal arc, and an arc smaller than a semicircle is called a suboptimal arc, so a semicircle is neither an optimal arc nor a suboptimal arc. The optimal arc is generally represented by three letters, and the suboptimal arc is generally represented by two letters. The optimal arc is an arc with a central angle greater than 180 degrees, and the suboptimal arc is an arc with a central angle less than 180 degrees.
In the same circle or equal circle, two arcs that can overlap each other are called equal arcs.
4) Angle
The angle of the vertex on the center of the circle is called the central angle.
The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle. The angle of a circle is equal to half the central angle of the same arc.
How can I learn math well?
1, be diligent
You can't just think with your head when learning mathematics. You must work hard when learning mathematics, because there are many times when we don't understand, but we may write thank you by hand.
Homework is very important.
An important way to learn mathematics is to finish the homework assigned by the teacher. It is not enough to just attend classes. Colleagues who finish the homework assigned by the teacher should do more exercises to consolidate after class.
3. Preview in class and review after class
Preview before class is very important, which allows us to concentrate the knowledge points that we didn't understand during preview and review them in time after class. After all, it's easy to forget when we listen in class without consolidation.
4. Summarize the wrong question bank
When studying mathematics, we can use our notebooks to record our mistakes and go back and do them again every three days or so. Some wrong questions may have been done at that time, but they may have been forgotten in a few days.
5, don't care too much about the problem.
When learning mathematics, we will encounter many kinds of problems, and sometimes teachers may not be able to solve them. At this time, we don't have to care too much. We will concentrate on understanding and doing basic questions, and most of them are still basic questions during the exam!