半角公式

三角学是处理角度、尺寸和度量的数学分支。当两条线相对于彼此倾斜时形成一个角度。角度的倾角称为测量，当几个角度聚集在一起时，它们会产生一个维度。

有几个公式和恒等式有助于确定倾角和测量值。确定 sin、cos、tan、cossec、sec 和 cot 的三角函数 0°、30°、45°、60°、90°、180° 的值。同样，我们在数学中也有一个叫做半角公式的东西。

半角公式

用于查找除了众所周知的 0°、30°、45°、60°、90°、180° 值之外的角度值。半角是从双角公式推导出来的，下面列出了 sin、cos 和 tan：

1. sin x/2 = +/- ((1 – cos x)/ 2) ^1/2
2. cos x/2 = +/- ((1 + cos x)/ 2) ^1/2
3. tan x/ 2 = (1 – cosx)/ sinx

双角公式的一些更重要的恒等式将有助于推导半角公式，

cos 2x = cos (x +x) 

cos 2x = (cosx)(cosx) – (sinx)(sinx)

cos 2x = cos²x – sin²x

cos 2x = cos²x – (1 -cos²x)

cos2x = 2cos²x – 1 ⇢ (eq. 1)

Now, put cos²x = 1 – sin²x

cos2x = 2 (1 – sin²x) – 1 

cos2x = 2 – 2sin²x- 1 

cos2x = 1 – 2sin²x ⇢ (eq. 2)

cos半角公式的推导

From above, we will make use of cos2x = 2cos²x – 1, equation 1 denoted by eq. 1,

Put x = 2y

cos (2)(y/2) = 2cos²(y/2) – 1

cos y = 2cos²(y/2) – 1

1 + cos y = 2cos²(y/2) 

Or 

2cos²(y/2) = 1 + cosy

cos²(y/2) = (1+ cosy)/2

cos(y/2) = +/- √(1+ cosy)/2

sin半角公式的推导

From above, we will make use of cos2x = 1 – 2sin²x , equation 2 denoted by eq2.

Put x = 2y

cos (2)(y/2) = 1 – 2sin²(y/2)     

cos y = 1 – 2sin²(y/2)   

2sin²(y/2) = 1 – cosy

sin²(y/2) = (1 – cosy)/2

sin(y/2) = +/- √(1 – cosy)/2

tan半角公式的推导

tan(x/2) = sin(x/2) / cos(x/2)

Putting the values of half angle for sin and cos. We get,

tan(x/2) = +/- ((√(1 – cosy)/2 ) / (√(1+ cosy)/2 ))

tan(x/2) = +/- (√(1 – cosy)/(1+ cosy) )

Rationalising the denominator

tan(x/2) = +/- (√(1 – cosy)(1 – cosy)/(1+ cosy)(1 – cosy))

tan(x/2) = +/- (√(1 – cosy)²/(1 – cos²y))

tan(x/2) = +/- (√(1 – cosy)²/( sin²y))

tan(x/2) = (1 – cosy)/( siny)

示例问题

问题一：确定sin 15°的值

解决方案：

We know that the formula for half angle of sine is given by:

sin x/2 = +/- ((1 – cos x)/ 2) ^1/2

The value of sine 15° can be found by substituting x as 30° in the above formula

sin 30°/2 = +/- ((1 – cos 30°)/ 2) ^1/2

sin 15° = +/- ((1 – 0.866)/ 2) ^1/2

sin 15° = +/- (0.134/ 2) ^1/2

sin 15° = +/- (0.067) ^1/2

sin 15° = +/- 0.2588

问题2：确定sin 22.5°的值

解决方案：

We know that the formula for half angle of sine is given by:

sin x/2 = +/- ((1 – cos x)/ 2) ^1/2

The value of sine 15° can be found by substituting x as 45° in the above formula

sin 45°/2 = +/- ((1 – cos 45°)/ 2) ^1/2

sin 22.5° = +/- ((1 – 0.707)/ 2) ^1/2

sin 22.5° = +/- (0.293/ 2) ^1/2

sin 22.5° = +/- (0.146) ^1/2

sin 22.5° = +/- 0.382

问题3：确定tan 15°的值

解决方案：

We know that the formula for half angle of sine is given by:

tan x/2 = +/- (1 – cos x)/ sin x

The value of tan 15° can be found by substituting x as 30° in the above formula

tan 30°/2 = +/- (1 – cos 30°)/ sin 30°

tan 15° = +/- (1 – 0.866)/ sin 30

tan 15° = +/- (0.134)/ 0.5

tan 15° = +/- 0.268

问题四：确定tan 22.5°的值

解决方案：

We know that the formula for half angle of sine is given by:

tan x/2 = +/- (1 – cos x)/ sin x

The value of tan 22.5° can be found by substituting x as 45° in the above formula

tan 30°/2 = +/- (1 – cos 45°)/ sin 45°

tan 22.5° = +/- (1 – 0.707)/ sin 45°

tan 22.5° = +/- (0.293)/ 0.707

tan 22.5° = +/- 0.414

问题5：确定cos 15°的值

解决方案：

We know that the formula for half angle of sine is given by:

cos x/2 = +/- ((1 + cos x)/ 2) ^1/2

The value of sine 15° can be found by substituting x as 30° in the above formula

cos 30°/2 = +/- ((1 + cos 30°)/ 2) ^1/2

cos 15° = +/- ((1 + 0.866)/ 2) ^1/2

cos 15° = +/- (1.866/ 2) ^1/2

cos 15° = +/- (0.933) ^1/2

cos 15° = +/- 0.965

问题6：确定cos 22.5°的值

解决方案：

We know that the formula for half angle of sine is given by:

cos x/2 = +/- ((1 + cos x)/ 2) ^1/2

The value of sine 15° can be found by substituting x as 45° in the above formula

cos 45°/2 = +/- ((1 + cos 45°)/ 2) ^1/2

cos 22.5° = +/- ((1 + 0.707)/ 2) ^1/2

cos 22.5° = +/- (1.707/ 2) ^1/2

cos 22.5° = +/- ( 0.853 ) ^1/2

cos 22.5° = +/- 0.923