找到 csc 225° 的准确值
三角学是一门数学学科，研究直角三角形的边长和角之间的关系。三角函数，也称为测角函数、角函数或圆函数，是建立角度与直角三角形两条边之比之间关系的函数。六个主要的三角函数是正弦、余弦、正切、余切、正割或余割。
Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.
如上图中的直角三角形所示：
- 斜边：与直角相对的边是斜边，它是直角三角形中最长的边，与90°角相对。
- 底：角 C 所在的一侧称为底。
- 垂直：考虑角度 C 的对边。
三角函数
三角函数有 6 个基本的三角函数，它们是正弦、余弦、正切、余割、正割和余切。现在让我们看看三角函数。六个三角函数如下，
- 正弦：它被定义为垂直和斜边的比率，它表示为 sin θ
- 余弦：定义为底边与斜边的比值，表示为 cos θ
- 正切：它被定义为一个角度的正弦和余弦之比。因此，切线的定义是垂直与底的比值，并表示为 tan θ
- cosecant：它是 sin θ 的倒数，表示为 cosec θ。
- 割线：它是 cos θ 的倒数，表示为 sec θ。
- cotangent：它是 tan θ 的倒数，表示为 cot θ。
根据上图，三角比是
Sin θ = Perpendicular / Hypotenuse = AB/AC
Cosine θ = Base / Hypotenuse = BC / AC
Tangent θ = Perpendicular / Base = AB / BC
Cosecant θ = Hypotenuse / Perpendicular = AC/AB
Secant θ = Hypotenuse / Base = AC/BC
Cotangent θ = Base / Perpendicular = BC/AB
互惠身份
Sin θ = 1/ Cosec θ OR Cosec θ = 1/ Sin θ
Cos θ = 1/ Sec θ OR Sec θ = 1 / Cos θ
Cot θ = 1 / Tan θ OR Tan θ = 1 / Cot θ
Cot θ = Cos θ / Sin θ OR Tan θ = Sin θ / Cos θ
Tan θ.Cot θ = 1
三角比值 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 Not Defined Sec θ Not Defined 2 √2 2/√3 1 Cosec θ 1 2/√3 √2 2 Not Defined Cot θ Not Defined √3 1 1/√3 0
补角和补角的三角恒等式
- 互补角：和等于90°的一对角
- 补角：和等于 180° 的一对角
互补角的恒等式是
sin (90° – θ) = cos θ
cos (90° – θ) = sin θ
tan (90° – θ) = cot θ
cot (90° – θ) = tan θ
sec (90° – θ) = cosec θ
cosec (90° – θ) = sec θ
补角的恒等式
sin (180° – θ) = sin θ
cos (180° – θ) = – cos θ
tan (180° – θ) = – tan θ
cot (180° – θ) = – cot θ
sec (180° – θ) = – sec θ
cosec (180° – θ) = – cosec θ
三角学象限
象限
csc 225°的准确值是多少？
解决方案：
Here cosec is positive only in 1st and 2nd Quadrant.
225° does not lies in 1st and 2nd Quadrant.
cosec is negative in 3rd Quadrant.
Therefore cosec (180° + θ) = – cosec θ
225° lies in 3rd Quadrant.
cosec (225°) = cosec (180° + 45°)
cosec (225°) = – cosec (45°)
cosec (225°) = – √2 (Since, cosec(45°) = √2)
Therefore, the value of cosec (225°) is -√2.
类似问题
问题 1：sin 270 的确切值是多少？
解决方案：
Here sin is positive only in 1st and 2nd Quadrant.
270° does not lies in 1st and 2nd Quadrant.
Therefore sin (360° – θ) = – sin θ
sin (270°) = sin (360° – 90°)
sin (270°) = – sin (90°)
sin (270°) = – 1
So the exact value of sin 270 is -1
问题2：cos 270的确切值是多少？
解决方案：
Here cos is positive only in 1st and 4th Quadrant.
270° lies in 3rd Quadrant.
Therefore cos(360° – θ) = – cos θ
cos(270°) = cos(360° – 90°)
cos(270°) = -cos(90°)
cos (270°) = 0 { as per the trigonometry value table }
So the exact value of cos 270 is 0.