什么是正弦定律?
几何是数学的一个分支,它关注形状、大小、它们的参数、测量、属性以及点和线之间的关系的研究。对于平面上表示的二维形状,存在三种几何类型。它们是欧几里得几何、球面几何和双曲几何。
给定的文章讨论了三角学的子主题。文章的内容包括对三角比的解释、不同角度的三角比的表格表示、对正弦定律及其数学推导的解释以及一些示例问题,以便更好地理解。
三角函数和三角比
三角学是数学的一个分支,它处理直角三角形的边和角的评估。三角运算是针对给定直角三角形的边和角以及一些标准三角比进行的。
三角比是基于直角三角形的边比的三角函数的数值。这些比率所依赖的直角三角形的三个边是斜边、垂边和底边。六个三角比是
- 正弦 (sin)
- 余弦 (cos)
- 切线(棕褐色)
- 余切 (cot)
- 割线 (cosec)
- 正割(秒)
三角比表
在三角运算中,一些特定值通常用于角度“θ”。三角比表包括角度为 0°、30°、45°、60° 和 90° 的三角比值。Angles 0° 30° 45° 60° 90° sin θ 0 1/2 1/√2 √3/2 1 cos θ 1 √3/2 1/√2 1/2 0 tan θ 0 1/√3 1 √3 ∞ cot θ ∞ √3 1 1/√3 0 sec θ 1 2√3 √2 2 ∞ cosec θ ∞ 2 √2 2√3 1
什么是正弦定律?
回答:
The Law of sines is also widely known as sine law, sine rule, and sine formula. The Law of sines is defined as the ratio of sides of a triangle and their equivalent or opposite sine angles. It is used to determine the unknown side or range of an oblique triangle. For the calculation using the Law of Sines at least, two angles are needed,
The other formula derived with respect to the law of sine is,
- a:b:c = sinA : sinB : sinC
- a/b = sinA/sinB
- b/c = sinB/sinC
Derivation of law of sines
Here, a right-angled triangle is required to prove the given trigonometric function.
Let ABC be a triangle with sides AB = c, BC = a, and AC = b.
Now, draw a perpendicular, CD perpendicular to AB. Then, CD = h which is the height of the triangle.
The perpendicular CD divided \triangleABC into two right-angled triangles CDA and CDB.
To demonstrate: a/b = sinA/sinB
In the CDA,
sinA = h/b
And, in triangle CDB,
sinB = h/a
Therefore,
sinA/sinB = (h/b) / (h/a)
sinA/sinB = a/b
Hence,proved.
Similarly, sinB/sinC = b/c canbe proved and any other pair of angles with respect to the opposite sides.
示例问题
问题1:角度的正弦是什么意思?
回答:
Sine of a angle is defined as a ratio of perpendicular side which stands opposite to thew angle of hypotenuse.
问题2:为什么要使用正弦定理?
回答:
Law of sines is used to determine the unknown angle or side of a triangle when other two angles and sides are given.
问题 3:如果 a、b 和 c 是边且 A、 B和 C 是三角形的角,余弦规则是什么?
回答:
If a, b and c are the sides and A, B and C are the angles of a triangle, then cosine rule is given by:
- a2 = b2 + c2 – 2bc cos A
- b2 = a2 + c2 – 2ac cos B
- c2 = a2 + b2 – 2ab cos C
问题4:三个基本三角函数是什么?
回答:
The three basic trigonometric functions are sine, cosine and tangent.