QUESTION 44.1 Using Exact Values And Without A Calculator, Simplify The Following Expression:${ \frac{\sin 330^{\circ} \cdot \tan 120^{\circ} \cdot \cos 240^{\circ}}{\tan 210^{\circ} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ}} }$4.2 Using

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on simplifying trigonometric expressions using exact values and without the aid of a calculator.

Understanding Trigonometric Functions

Before we dive into simplifying trigonometric expressions, it is essential to understand the basic trigonometric functions. These functions are:

  • Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite an angle to the length of the side adjacent to an angle.

Simplifying the Given Expression

The given expression is:

sin⁡330∘⋅tan⁡120∘⋅cos⁡240∘tan⁡210∘⋅cos⁡120∘⋅sin⁡150∘{ \frac{\sin 330^{\circ} \cdot \tan 120^{\circ} \cdot \cos 240^{\circ}}{\tan 210^{\circ} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ}} }

To simplify this expression, we need to use the trigonometric identities and the properties of the trigonometric functions.

Step 1: Simplify the Numerator

The numerator of the expression is:

sin⁡330∘⋅tan⁡120∘⋅cos⁡240∘{ \sin 330^{\circ} \cdot \tan 120^{\circ} \cdot \cos 240^{\circ} }

We can simplify this expression by using the trigonometric identities:

  • sin (A + B) = sin A cos B + cos A sin B
  • cos (A + B) = cos A cos B - sin A sin B
  • tan A = sin A / cos A

Using these identities, we can rewrite the numerator as:

sin⁡330∘⋅tan⁡120∘⋅cos⁡240∘=sin⁡(180∘+150∘)⋅tan⁡120∘⋅cos⁡(180∘+60∘){ \sin 330^{\circ} \cdot \tan 120^{\circ} \cdot \cos 240^{\circ} = \sin (180^{\circ} + 150^{\circ}) \cdot \tan 120^{\circ} \cdot \cos (180^{\circ} + 60^{\circ}) }

Simplifying further, we get:

sin⁡330∘⋅tan⁡120∘⋅cos⁡240∘=−sin⁡150∘⋅tan⁡120∘⋅−cos⁡60∘{ \sin 330^{\circ} \cdot \tan 120^{\circ} \cdot \cos 240^{\circ} = -\sin 150^{\circ} \cdot \tan 120^{\circ} \cdot -\cos 60^{\circ} }

Using the trigonometric identity tan A = sin A / cos A, we can rewrite the expression as:

−sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅−cos⁡60∘{ -\sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot -\cos 60^{\circ} }

Simplifying further, we get:

sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅cos⁡60∘{ \sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot \cos 60^{\circ} }

Step 2: Simplify the Denominator

The denominator of the expression is:

tan⁡210∘⋅cos⁡120∘⋅sin⁡150∘{ \tan 210^{\circ} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ} }

We can simplify this expression by using the trigonometric identities:

  • tan A = sin A / cos A
  • cos (A + B) = cos A cos B - sin A sin B
  • sin (A + B) = sin A cos B + cos A sin B

Using these identities, we can rewrite the denominator as:

tan⁡210∘⋅cos⁡120∘⋅sin⁡150∘=sin⁡210∘cos⁡210∘⋅cos⁡120∘⋅sin⁡150∘{ \tan 210^{\circ} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ} = \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ} }

Simplifying further, we get:

sin⁡210∘cos⁡210∘⋅cos⁡120∘⋅sin⁡150∘=sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘{ \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot \cos 120^{\circ} \cdot \sin 150^{\circ} = \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} }

Using the trigonometric identity sin (A + B) = sin A cos B + cos A sin B, we can rewrite the expression as:

sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘=sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅(sin⁡150∘+cos⁡150∘sin⁡60∘){ \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} = \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot (\sin 150^{\circ} + \cos 150^{\circ} \sin 60^{\circ}) }

Simplifying further, we get:

sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅(sin⁡150∘+cos⁡150∘sin⁡60∘)=sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘{ \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot (\sin 150^{\circ} + \cos 150^{\circ} \sin 60^{\circ}) = \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ} }

Step 3: Simplify the Expression

Now that we have simplified the numerator and the denominator, we can simplify the expression by dividing the numerator by the denominator.

sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅cos⁡60∘sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘{ \frac{\sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot \cos 60^{\circ}}{\frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ}} }

Simplifying further, we get:

sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅cos⁡60∘sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘=sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅cos⁡60∘sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘{ \frac{\sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot \cos 60^{\circ}}{\frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ}} = \frac{\sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot \cos 60^{\circ}}{\frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ}} }

Using the trigonometric identity tan A = sin A / cos A, we can rewrite the expression as:

sin⁡150∘⋅sin⁡120∘cos⁡120∘⋅cos⁡60∘sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−sin⁡210∘cos⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘=sin⁡150∘⋅tan⁡120∘⋅cos⁡60∘tan⁡210∘⋅−cos⁡60∘⋅sin⁡150∘−tan⁡210∘⋅−cos⁡60∘⋅cos⁡150∘sin⁡60∘{ \frac{\sin 150^{\circ} \cdot \frac{\sin 120^{\circ}}{\cos 120^{\circ}} \cdot \cos 60^{\circ}}{\frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \frac{\sin 210^{\circ}}{\cos 210^{\circ}} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ}} = \frac{\sin 150^{\circ} \cdot \tan 120^{\circ} \cdot \cos 60^{\circ}}{\tan 210^{\circ} \cdot -\cos 60^{\circ} \cdot \sin 150^{\circ} - \tan 210^{\circ} \cdot -\cos 60^{\circ} \cdot \cos 150^{\circ} \sin 60^{\circ}} }

Simplifying further, we get:

Q&A: Simplifying Trigonometric Expressions

Q: What is the main goal of simplifying trigonometric expressions? A: The main goal of simplifying trigonometric expressions is to reduce complex expressions into simpler forms, making it easier to work with them.

Q: What are some common trigonometric identities used to simplify expressions? A: Some common trigonometric identities used to simplify expressions include:

  • sin (A + B) = sin A cos B + cos A sin B
  • cos (A + B) = cos A cos B - sin A sin B
  • tan A = sin A / cos A

Q: How do you simplify a trigonometric expression with multiple terms? A: To simplify a trigonometric expression with multiple terms, you can use the following steps:

  1. Combine like terms: Combine terms that have the same trigonometric function.
  2. Use trigonometric identities: Use trigonometric identities to simplify each term.
  3. Simplify the expression: Simplify the expression by combining the simplified terms.

Q: What is the difference between a trigonometric expression and a trigonometric function? A: A trigonometric expression is a combination of trigonometric functions, while a trigonometric function is a single trigonometric function.

Q: How do you simplify a trigonometric expression with a fraction? A: To simplify a trigonometric expression with a fraction, you can use the following steps:

  1. Simplify the numerator and denominator: Simplify the numerator and denominator separately.
  2. Use trigonometric identities: Use trigonometric identities to simplify the fraction.
  3. Simplify the expression: Simplify the expression by combining the simplified numerator and denominator.

Q: What is the importance of simplifying trigonometric expressions? A: Simplifying trigonometric expressions is important because it makes it easier to work with complex expressions, which can be used to solve problems in various fields, including physics, engineering, and navigation.

Q: How do you know when to simplify a trigonometric expression? A: You should simplify a trigonometric expression when:

  • The expression is complex: The expression is complex and difficult to work with.
  • The expression is used in a problem: The expression is used in a problem and needs to be simplified to solve the problem.
  • The expression is used in a formula: The expression is used in a formula and needs to be simplified to derive the formula.

Conclusion

Simplifying trigonometric expressions is an essential skill in mathematics and has numerous applications in various fields. By understanding the basic trigonometric functions and using trigonometric identities, you can simplify complex expressions and make it easier to work with them. Remember to simplify expressions when they are complex, used in a problem, or used in a formula.

Frequently Asked Questions

  • What is the difference between a trigonometric expression and a trigonometric function? A trigonometric expression is a combination of trigonometric functions, while a trigonometric function is a single trigonometric function.
  • How do you simplify a trigonometric expression with multiple terms? To simplify a trigonometric expression with multiple terms, you can use the following steps: combine like terms, use trigonometric identities, and simplify the expression.
  • What is the importance of simplifying trigonometric expressions? Simplifying trigonometric expressions is important because it makes it easier to work with complex expressions, which can be used to solve problems in various fields.

Glossary

  • Trigonometric expression: A combination of trigonometric functions.
  • Trigonometric function: A single trigonometric function.
  • Trigonometric identity: A mathematical statement that describes the relationship between trigonometric functions.
  • Simplifying trigonometric expressions: The process of reducing complex trigonometric expressions into simpler forms.