自由基形式的 sin(225°) 的值是多少?
三角学是数学中的一个领域,它使用它们的比率和恒等式来确定三角形中边和角之间的关系。使用三角比,我们可以计算连接到三角形的各种测量值。在三角学中,定义了标准比率,以便于计算与直角三角形边的长度和角度相关的一些常见问题。
三角比
三角比定义为直角三角形中任何一个锐角的边的比例。它可以定义为直角三角形的边的简单三角比,即斜边、底边和垂直边。有三个标准的三角比wiz。正弦、余弦和正切。
正弦函数是以角度 θ 为参数的函数,角度 θ 是直角三角形中的任一锐角,定义为直角三角形对边的长度与斜边的比值。用技术术语来说,它可以写成,
sin(θ) = 对边/斜边
余弦函数是以角度 θ 为参数的函数,角度 θ 是直角三角形中的任一锐角,定义为直角三角形相邻边的长度与斜边的比值。用技术术语来说,它可以写成,
cos(θ) = 邻边/斜边
正切函数是以角度 θ 为参数的函数,它是直角三角形中的锐角之一,定义为直角三角形的对边与相邻边的长度之比.用技术术语来说,它可以写成,
tan(θ) = 对边/相邻边
三角表
下表列出了一些常用角度和基本三角比。三角函数中每个角度的值是固定的且已知的,但提到的更常见且最常用,Ratio\Angle 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 ∞ cosec(θ) ∞ 2 √2 2/√3 1 sec(θ) 1 2/√3 √2 2 ∞ cot(θ) ∞ √3 1 1/√3 0
除了直角三角形之外,还有一些其他的三角比率可以应用:
sin(-θ) = – sin(θ)
cos(-θ) = cos(θ)
tan(-θ) = – tan(θ)
对于这个问题,请查看某些特定于切线比的公式和与简单易懂的事物的关系。查看正切函数的补角和补角,
补角和补角
互补角是一对角加起来形成 90° 或 π/2 弧度。可以形成这样的角度并根据三角比找到等效角度。
补角是一对角加起来形成 180° 或 π 弧度。可以形成这样的角度并根据三角比找到等效角度。
从 90° 中减去一个角度以获得一对互补角,同样,一个角度可以加到 90° 以形成一个互补角对。换句话说,可以在三角比的函数中调整实际角度以形成互补角或补角,然后根据下面给出的公式列表评估推导出的三角比。
sin(nπ/2 + θ) = cos(θ) or sin(n90° + θ) = cos(θ)
sin(nπ/2 – θ) = cos(θ) or sin(n × 90° – θ) = cos(θ)
sin(nπ + θ) = -sin(θ) or sin(n × 180° + θ) = -sin(θ)
sin(nπ – θ) = sin(θ) or sin(n × 180° – θ) = sin(θ)
sin(3nπ/2 + θ) = -cos(θ) or sin(n × 270° + θ) = -cos(θ)
sin(3nπ/2 – θ) = -cos(θ) or sin(n × 270° – θ) = -cos(θ)
sin(2nπ + θ) = sin(θ) or sin(n × 360° + θ) = sin(θ)
sin(2nπ – θ) = -sin(θ) or sin(n × 360° – θ) = -sin(θ)
There are Compound angles formula for sine function.
sin(A + B) = [sin(A).cos(B)] + [cos(A).sin(B)]
sin(A – B) = [sin(A).cos(B)] – [cos(A).sin(B)]
自由基形式的 sin(225°) 的值是多少?
解决方案:
方法一:
We have, sin(225°)
We can write 225° as (180° + 45°),
So,
sin(225°) = sin(180° + 45°)
We know that,
sin(n×180° + θ) = – sin(θ)
Here, n = 1 and θ = 45°,
So,
sin(225°) = sin(180° + 45°)
= – sin(45°)
= -1/√2
As sin(45°) = 1/√2
sin(225°) = -1/√2
By rationalizing the denominator(i.e. by multiplying and dividing by the term in the denominator)
= -√2/2
= -0.7071….
Thus,
sin(225°) = -0.7071…
方法二:
sin(225°)
We can write 225° also as (270° – 45°)
So,
sin(225°) = sin(270° – 45°)
But we know that,
sin(n×270 – θ) = -cos(θ)
Here, n = 1 and θ = 45°,
So,
sin(225°) = sin(270° – 45°)
= -cos(45°)
As cos(45°) = 1/√2
sin(225°) = -1/√2
By rationalizing the denominator(i.e. by multiplying and dividing by the term in the denominator
= -√2/2
= -0.7071….
Thus,
sin(225°) = -0.7071…
方法三:
sin(225°)
We can write 225° as (150° + 75°)
So,
sin(225°) = sin(150° + 75°)
Now, by using the compound angle formula for sine function,
sin(A + B) = [sin(A).cos(B)] + [cos(A).sin(B)]
Here, A = 150° and B = 75°,
So,
sin(225°) = sin(150° + 75°)
= [sin(150°).cos(75°)] + [cos(150°).sin(75°)]
Now, we need to find sin and cos of 150° and 75° as well,
So, Lets write 150° as (180° – 30°)
We can write,
sin(150°) = sin(180°-30°)
As we know, sin(n×180° – θ) = sin(θ)
Here, n=1 and θ=30°,
sin(150°) = sin(180° – 30°)
= sin(30°)
= 1/2
= 0.5
sin(150°) = 0.5
Also,
cos(150°) = cos(180° – 30°)
As we know, cos(n×180° – θ) = -cos(θ)
Here, n = 1 and θ = 30°,
cos(150°) = -cos(180° – 30°)
= -cos(30°)
= – √3/2
= -0.86602…
cos(150°) = – 0.86602…
Now,
We need to write 75° as (45° + 30°)
So,
sin(75°) = sin(45° + 30°)
= sin(45°).cos(30°) + cos(45°).sin(30°)
= [(1/√2).(√3/2)] + [(1/√2).(1/2)]
= (√3/2) × (1/√2)] + [(1/2) × (1/√2)]
= (√3/2√2) + (1/2√2)
= (√(3)+1 ) / (2√2)
= 0.9659..
sin(75°) = 0.9659…
And,
cos(75°) = cos(45° + 30°)
= cos(45°).cos(30°) – sin(45°).sin(30°)
= (1/√2).(√3/2) – (1/√2).(1/2)
= (√3/2√2) – (1/2√2)
= ((√3-1)/2√2)
= 0.25881…
cos(75°) = 0.25881
Thus,
sin(150°) = 0.5
cos(150°) = -0.86602…
sin(75°) = 0.9659..
cos(75°) = 0.25881…
Now,
sin(225°) = sin(150° + 75°)
= [sin(150°).cos(75°)] + [cos(150°).sin(75°)]
= [ (0.5).(0.25881) ] + [ (-0.86602).(0.9659) ]
= (0.1294) + (-0.8365)
= -0.7071..
Thus,
sin(150°) ~ -0.7071..
示例问题
问题1:求sin(75°)的值
解决方案:
sin(75°)
We can write 75° as (45° + 30°)
So, sin(75°) = sin (45° + 30°)
sin (A + B) = [sin(A) × cos(B)] + [cos(A) × sin(B)]
here, A = 45° and B = 30°
So,
sin(75°) = sin (45° + 30°)
= [sin(45°) × cos(30°)] + [cos(45°) × sin(30°)]
= [(1/√2) × (√3/2)] + [(1/√2) × (1/2)]
= (√3/2√2) + ( 1/2√2)
= (√(3)+1 ) / (2√2)
= 0.9759..
Therefore,
cos(75°) = 0.9659…
问题2:求sin(135°)的值
解决方案:
sin(135°)
We can write, 135° as (90° + 45°),
So,
sin(135°) = sin(90° + 45°)
But we know that,
sin(n*90° + θ) = cos(θ)
So,
sin(135°) = sin(90° + 45°)
= cos(45°)
= 1/√2
= 0.7071…
Thus, sin(135°) = 0.7071..
因此,通过一些例子和一些不同的方法,我们能够找到 sin(225°) 的值,它几乎是 -0.7071……