什么是 Cos Square theta 公式？

将任何变量的不同三角函数关联起来的方程称为三角恒等式。这些三角恒等式帮助我们将各种三角公式和关系与不同的角度联系起来。它们是正弦、余弦、正切、余切、秒和余弦。在这里，我们将看看cos square theta 公式。

根据三角恒等式，cos平方theta公式由下式给出

cos²θ + sin²θ = 1

where θ is an acute angle of a right-angled triangle.

证明：

The trigonometric functions for any right angled triangle is defined as:

cosθ = base/hypotenuse

sinθ = altitude/hypotenuse

So, we can write

cos²θ + sin²θ = base²/hypotenuse² + altitude²/hypotenuse²

Thus, cos²θ + sin²θ = (base² + altitude²)/hypotenuse²

Applying pyhogorus theorem for right angled triangle, we get

base² + altitude² = hypotenuse²

Thus, we get

cos²θ + sin²θ = 1

除此之外，还有一些使用此属性导出的通用公式：

• cos ² θ = 1 – sin ² θ
• cos2θ = cos ² θ – sin ² θ
• cos2θ = 2cos ² θ – 1

示例问题

问题 1. 给定 sinθ 的值，求 cosθ 的值是 3/5。

解决方案：

Given, the value of sinθ = 3/5

Using cos square formula, we get

cos²θ + sin²θ = 1

cos²θ = 1 – sin²θ = 1 – (3/5)² = 1 – 9/25

cos²θ = 16/25

cosθ = √(16/25) = ± 4/5

Thus, the value of cosθ is ± 4/5.

问题 2. 求 cosθ 的值，给定 cosθ – sinθ = 1

解决方案：

Given, cosθ – sinθ = 1.

or, cosθ = 1 + sinθ        —- (i)

Using cos square formula, we get

cos²θ + sin²θ = 1

cos²θ = 1 – sin²θ = (1 + sinθ)(1 – sinθ)

cos²θ = cosθ (1 – sinθ)

cosθ (cosθ – 1 + sinθ) = 0

So, we get two cases,

cosθ = 0

else, cosθ – 1 + sinθ = 0

or, cosθ = 1 – sinθ     —- (ii)

From eq.(i) and eq.(ii), we get

1 – sinθ = 1 + sinθ

2sinθ = 0

sinθ = 0

From eq.(i), we get cosθ = 1 + sinθ = 1 + 0 = 1

Thus, cosθ = 1

So, we get two possibilities. The value of cosθ is 0 or 1.

问题 3. 如果 cosθ = 3/5，求 sin ² θ – cos ² θ 的值。

解决方案：

Given, the value of cosθ = 3/5

Now, using cos square formula, we can write

sin²θ – cos²θ = (1 – cos²θ) – cos²θ = 1 – 2cos²θ

Putting the value of cosθ = 3/5, we get

sin²θ – cos²θ = 1 – 2cos²θ = 1 – 2 × (3/5)² = 1 – 2 × 9/25 =1 – 18/25 = 7/25

So, the answer is 7/25 .

问题 4. 给定 cosθ = 1/2，求 cos2θ 的值。

解决方案：

Using the generalized formula,

cos2θ = 2cos²θ – 1

we can find the value of cos2θ, by substituting cosθ = 1/2

cos2θ = 2 × (1/2)² – 1 = 2/4 – 1 = 1/2 – 1 = – 1/2

Thus, cos2θ = – 1/2.