如果 A 和 B 是锐角，使得 cosA = cosB，则证明 A = B
三角学基本上是研究三角形的角度和边之间的关系。它是日常生活中广泛使用的数学主题之一。它涉及对直角三角形的操作，即一个角度等于 90° 的三角形。在继续之前，我们应该了解一些术语。这些条款是，
- 斜边 – 它是直角三角形中与直角相对的一侧。它是直角三角形的最长边。在图 1 中，AC 侧是斜边。
- 垂直 - 三角形的垂线，对应于一个特别锐角 θ 是角度 θ 的对边。在图 1 中，边 AB 是对应于角度 θ 的垂线。
- 底 - 它是与特别锐角 θ 相邻的一侧。在图 1 中，边 BC 是对应于角度 θ 的底边。
图1
如前所述，三角学描述了直角三角形的角和边之间的关系。这种关系由标准比率表示，并给出如下
- 正弦 (sin) – 角 θ 的正弦是对应于角 θ 的垂线长度与三角形斜边长度之比。
sinθ = 垂直/斜边 = p/h
- 余弦 (cos) – 角 θ 的余弦是对应于角 θ 的底边长度与三角形斜边长度之比。
cosθ = 底边/斜边 = b/h
- 正切 (tan) – 角度 θ 的正切是对应于角度 θ 的垂线长度与三角形特定角度的底边长度之比。
cosθ = 底边/斜边 = b/h
- 余切（cot） ——它是正切的倒数。
cotθ = 1/tanθ = 底/垂直 = b/p
- 割线 (sec) – 它是余弦的倒数。
secθ = 1/cosθ = 斜边/底边 = h/b
- 余割 (cosec) – 它是正弦的倒数。
cosecθ = 1/sinθ = 斜边/垂直 = h/p
如果 A 和 B 是锐角，使得 cosA = cosB，则证明 A = B
解决方案：
Given,
cos A = cos B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Figure 2
Then cos A = AC/AB ⇢ (i)
cos B = BC/AB ⇢ (ii)
Since, cos A = cos B
Therefore,
AC/AB = BC/AB
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
类似问题
问题 1：如果 A 和 B 是锐角，使得 sin A = sin B，则检查 A = B。
解决方案：
Given,
sin A = sin B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Then sin A = BC/AB ⇢ (i)
sin B = AC/AB ⇢ (ii)
Since, sin A = sin B
Therefore,
BC/AB = AC/AB
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
问题 2：如果 A 和 B 是锐角，使得 tan A = tan B 则检查 A = B。
解决方案：
Given,
tan A = tan B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Then tan A = BC/AC ⇢ (i)
tan B = AC/BC ⇢ (ii)
Since, tan A = tan B
Therefore,
BC/AC = AC/BC
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
问题 3：如果 A 和 B 是锐角，使得 cot A = cot B 则检查 A = B。
解决方案：
Given,
cot A = cot B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Then cot A = AC/BC ⇢ (i)
cot B = BC/AC ⇢ (ii)
Since, cot A = cot B
Therefore,
AC/BC = BC/AC
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
问题 4：如果 A 和 B 是锐角，使得 sec A = sec B，则检查 A = B。
解决方案：
Given,
sec A = sec B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Then sec A = AB/AC ⇢ (i)
sec B = AB/BC ⇢ (ii)
Since, sec A = sec B
Therefore,
AB/AC = AB/BC
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
问题 5：如果 A 和 B 是锐角，且 cosec A = cosec B，则检查 A = B。
解决方案：
Given,
cosec A = cosec B
Suppose the triangle looks something like the figure 2, with acute angles A and B.
Then cosec A = AB/BC ⇢ (i)
cosec B = AB/AC ⇢ (ii)
Since, cosec A = cosec B
Therefore,
AB/BC = AB/AC
For the equation to be correct, AC should be equal to BC
=> angle A = angle B (since angles opposite to equal sides are equal)
所以，一般来说，我们可以说如果一个锐角的三角比等于另一个锐角的三角比，那么这些角度应该是相同的。