Given That $\alpha$ And $\beta$ Are Two Angles, Investigate Whether $\cos (\alpha - \beta)$ Is Distributive Under Subtraction And Derive Its Compound Angle Identity.1.1 Calculate $\cos(180^{\circ} -

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Investigating the Distributivity of Cosine Function under Subtraction and Deriving Compound Angle Identity

Introduction

In trigonometry, the cosine function is a fundamental concept that plays a crucial role in solving various mathematical problems. One of the essential properties of the cosine function is its distributivity under addition, which is expressed as cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos (a + b) = \cos a \cos b - \sin a \sin b. However, when it comes to subtraction, the distributivity of the cosine function is not as straightforward. In this article, we will investigate whether cos⁡(α−β)\cos (\alpha - \beta) is distributive under subtraction and derive its compound angle identity.

Distributivity of Cosine Function under Subtraction

To determine whether cos⁡(α−β)\cos (\alpha - \beta) is distributive under subtraction, we need to examine the expression cos⁡(α−β)\cos (\alpha - \beta). Using the angle subtraction formula, we can rewrite this expression as cos⁡αcos⁡β+sin⁡αsin⁡β\cos \alpha \cos \beta + \sin \alpha \sin \beta. This suggests that the cosine function may not be distributive under subtraction, as the expression does not resemble the distributive property of the cosine function under addition.

Deriving Compound Angle Identity

To derive the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta), we can use the angle subtraction formula. The angle subtraction formula states that cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos (a - b) = \cos a \cos b + \sin a \sin b. By substituting α\alpha for aa and β\beta for bb, we can rewrite the expression as cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.

Derivation of Compound Angle Identity

To derive the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta), we can use the following steps:

  1. Start with the angle subtraction formula: The angle subtraction formula states that cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos (a - b) = \cos a \cos b + \sin a \sin b.
  2. Substitute α\alpha for aa and β\beta for bb: By substituting α\alpha for aa and β\beta for bb, we can rewrite the expression as cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.
  3. Use the Pythagorean identity: The Pythagorean identity states that sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1. We can use this identity to rewrite the expression as cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β=cos⁡αcos⁡β+1−cos⁡2α1−cos⁡2β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta = \cos \alpha \cos \beta + \sqrt{1 - \cos^2 \alpha} \sqrt{1 - \cos^2 \beta}.

Conclusion

In conclusion, we have investigated whether cos⁡(α−β)\cos (\alpha - \beta) is distributive under subtraction and derived its compound angle identity. The compound angle identity for cos⁡(α−β)\cos (\alpha - \beta) is cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta. This identity can be used to simplify expressions involving the cosine function and subtraction.

Example 1: Calculating cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ})

To calculate cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ}), we can use the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta). By substituting α=180∘\alpha = 180^{\circ} and β=30∘\beta = 30^{\circ}, we can rewrite the expression as cos⁡(180∘−30∘)=cos⁡180∘cos⁡30∘+sin⁡180∘sin⁡30∘\cos(180^{\circ} - 30^{\circ}) = \cos 180^{\circ} \cos 30^{\circ} + \sin 180^{\circ} \sin 30^{\circ}.

Solution

Using the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta), we can calculate cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ}) as follows:

cos⁡(180∘−30∘)=cos⁡180∘cos⁡30∘+sin⁡180∘sin⁡30∘\cos(180^{\circ} - 30^{\circ}) = \cos 180^{\circ} \cos 30^{\circ} + \sin 180^{\circ} \sin 30^{\circ}

=(−1)(32)+(0)(12)= (-1) \left(\frac{\sqrt{3}}{2}\right) + (0) \left(\frac{1}{2}\right)

=−32= -\frac{\sqrt{3}}{2}

Therefore, cos⁡(180∘−30∘)=−32\cos(180^{\circ} - 30^{\circ}) = -\frac{\sqrt{3}}{2}.

Example 2: Calculating cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ})

To calculate cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ}), we can use the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta). By substituting α=90∘\alpha = 90^{\circ} and β=45∘\beta = 45^{\circ}, we can rewrite the expression as cos⁡(90∘−45∘)=cos⁡90∘cos⁡45∘+sin⁡90∘sin⁡45∘\cos(90^{\circ} - 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} + \sin 90^{\circ} \sin 45^{\circ}.

Solution

Using the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta), we can calculate cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ}) as follows:

cos⁡(90∘−45∘)=cos⁡90∘cos⁡45∘+sin⁡90∘sin⁡45∘\cos(90^{\circ} - 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} + \sin 90^{\circ} \sin 45^{\circ}

=(0)(22)+(1)(22)= (0) \left(\frac{\sqrt{2}}{2}\right) + (1) \left(\frac{\sqrt{2}}{2}\right)

=22= \frac{\sqrt{2}}{2}

Therefore, cos⁡(90∘−45∘)=22\cos(90^{\circ} - 45^{\circ}) = \frac{\sqrt{2}}{2}.

Conclusion

In conclusion, we have investigated whether cos⁡(α−β)\cos (\alpha - \beta) is distributive under subtraction and derived its compound angle identity. The compound angle identity for cos⁡(α−β)\cos (\alpha - \beta) is cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta. This identity can be used to simplify expressions involving the cosine function and subtraction. We have also provided examples of calculating cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ}) and cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ}) using the compound angle identity.
Frequently Asked Questions (FAQs) about the Distributivity of Cosine Function under Subtraction and Compound Angle Identity

Q: What is the distributivity of the cosine function under subtraction?

A: The distributivity of the cosine function under subtraction refers to the property of the cosine function that allows us to simplify expressions involving the cosine function and subtraction. Specifically, it states that cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.

Q: Why is the distributivity of the cosine function under subtraction important?

A: The distributivity of the cosine function under subtraction is important because it allows us to simplify expressions involving the cosine function and subtraction. This can be useful in a variety of mathematical applications, such as solving trigonometric equations and simplifying trigonometric expressions.

Q: How do I use the distributivity of the cosine function under subtraction to simplify expressions?

A: To use the distributivity of the cosine function under subtraction to simplify expressions, you can follow these steps:

  1. Identify the expression: Identify the expression that you want to simplify.
  2. Apply the distributivity formula: Apply the distributivity formula, which states that cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.
  3. Simplify the expression: Simplify the expression using the distributivity formula.

Q: What is the compound angle identity for cos⁡(α−β)\cos (\alpha - \beta)?

A: The compound angle identity for cos⁡(α−β)\cos (\alpha - \beta) is cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.

Q: How do I use the compound angle identity to simplify expressions?

A: To use the compound angle identity to simplify expressions, you can follow these steps:

  1. Identify the expression: Identify the expression that you want to simplify.
  2. Apply the compound angle identity: Apply the compound angle identity, which states that cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.
  3. Simplify the expression: Simplify the expression using the compound angle identity.

Q: What are some examples of using the distributivity of the cosine function under subtraction and compound angle identity?

A: Some examples of using the distributivity of the cosine function under subtraction and compound angle identity include:

  • Calculating cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ}): Using the compound angle identity, we can calculate cos⁡(180∘−30∘)\cos(180^{\circ} - 30^{\circ}) as follows: cos⁡(180∘−30∘)=cos⁡180∘cos⁡30∘+sin⁡180∘sin⁡30∘=(−1)(32)+(0)(12)=−32\cos(180^{\circ} - 30^{\circ}) = \cos 180^{\circ} \cos 30^{\circ} + \sin 180^{\circ} \sin 30^{\circ} = (-1) \left(\frac{\sqrt{3}}{2}\right) + (0) \left(\frac{1}{2}\right) = -\frac{\sqrt{3}}{2}.
  • Calculating cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ}): Using the compound angle identity, we can calculate cos⁡(90∘−45∘)\cos(90^{\circ} - 45^{\circ}) as follows: cos⁡(90∘−45∘)=cos⁡90∘cos⁡45∘+sin⁡90∘sin⁡45∘=(0)(22)+(1)(22)=22\cos(90^{\circ} - 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} + \sin 90^{\circ} \sin 45^{\circ} = (0) \left(\frac{\sqrt{2}}{2}\right) + (1) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}.

Q: What are some common mistakes to avoid when using the distributivity of the cosine function under subtraction and compound angle identity?

A: Some common mistakes to avoid when using the distributivity of the cosine function under subtraction and compound angle identity include:

  • Not applying the distributivity formula correctly: Make sure to apply the distributivity formula correctly, which states that cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.
  • Not simplifying the expression correctly: Make sure to simplify the expression correctly using the distributivity formula.
  • Not using the compound angle identity correctly: Make sure to use the compound angle identity correctly, which states that cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta.

Q: What are some real-world applications of the distributivity of the cosine function under subtraction and compound angle identity?

A: Some real-world applications of the distributivity of the cosine function under subtraction and compound angle identity include:

  • Solving trigonometric equations: The distributivity of the cosine function under subtraction and compound angle identity can be used to solve trigonometric equations.
  • Simplifying trigonometric expressions: The distributivity of the cosine function under subtraction and compound angle identity can be used to simplify trigonometric expressions.
  • Modeling real-world phenomena: The distributivity of the cosine function under subtraction and compound angle identity can be used to model real-world phenomena, such as the motion of objects in physics and engineering.