Using the Given Quantity to Express Trigonometric Functions

Oct 13, 2025 by windailys.com 139 views

Introduction

Trigonometric expressions are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will focus on writing trigonometric expressions in terms of a given quantity, specifically in terms of the sine function. We will use the given quantity $\sin 40^{\circ}=P$ to express the following trigonometric expressions: $0.1 \sin 140^{\circ}$ and $1.2 \sin 220^{\circ}$ .

Using the Given Quantity to Express Trigonometric Functions

To express the given trigonometric functions in terms of the given quantity $P$ , we need to use the trigonometric identities and the properties of the sine function. We will start by expressing the sine function in terms of the given quantity $P$ .

Expressing the Sine Function in Terms of P

We are given that $\sin 40^{\circ}=P$ . We can use this information to express the sine function in terms of $P$ . Since the sine function is periodic with a period of $360^{\circ}$ , we can write:

\sin (180^{\circ}+x)=\sin x

Using this identity, we can rewrite the given trigonometric functions in terms of $P$ .

Expressing $0.1 \sin 140^{\circ}$ in Terms of P

We can rewrite $0.1 \sin 140^{\circ}$ as:

0.1 \sin 140^{\circ}=0.1 \sin (180^{\circ}-40^{\circ})

Using the identity $\sin (180^{\circ}-x)=\sin x$ , we can rewrite this expression as:

0.1 \sin 140^{\circ}=0.1 \sin 40^{\circ}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

0.1 \sin 140^{\circ}=0.1 P

Expressing $1.2 \sin 220^{\circ}$ in Terms of P

We can rewrite $1.2 \sin 220^{\circ}$ as:

1.2 \sin 220^{\circ}=1.2 \sin (180^{\circ}+40^{\circ})

Using the identity $\sin (180^{\circ}+x)=-\sin x$ , we can rewrite this expression as:

1.2 \sin 220^{\circ}=-1.2 \sin 40^{\circ}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

1.2 \sin 220^{\circ}=-1.2 P

Conclusion

In this article, we have shown how to write trigonometric expressions in terms of a given quantity, specifically in terms of the sine function. We have used the given quantity $\sin 40^{\circ}=P$ to express the following trigonometric expressions: $0.1 \sin 140^{\circ}$ and $1.2 \sin 220^{\circ}$ . We have used the trigonometric identities and the properties of the sine function to rewrite these expressions in terms of $P$ . The resulting expressions are $0.1 P$ and $-1.2 P$ , respectively.

Applications of Writing Trigonometric Expressions in Terms of a Given Quantity

Writing trigonometric expressions in terms of a given quantity has many applications in various fields such as physics, engineering, and computer science. For example, in physics, we can use this technique to express the motion of an object in terms of its position, velocity, and acceleration. In engineering, we can use this technique to design and analyze mechanical systems, such as gears and linkages. In computer science, we can use this technique to develop algorithms for solving trigonometric equations.

Example Problems

Problem 1

Express $0.5 \sin 300^{\circ}$ in terms of $P$ .

Solution

We can rewrite $0.5 \sin 300^{\circ}$ as:

0.5 \sin 300^{\circ}=0.5 \sin (180^{\circ}+120^{\circ})

Using the identity $\sin (180^{\circ}+x)=-\sin x$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \sin 120^{\circ}

Since $\sin 120^{\circ}=\sin (180^{\circ}-60^{\circ})=\sin 60^{\circ}$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \sin 60^{\circ}

Since $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \frac{\sqrt{3}}{2}

Simplifying this expression, we get:

0.5 \sin 300^{\circ}=-\frac{\sqrt{3}}{4}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-\frac{\sqrt{3}}{4} P

Problem 2

Express $2.5 \sin 320^{\circ}$ in terms of $P$ .

Solution

We can rewrite $2.5 \sin 320^{\circ}$ as:

2.5 \sin 320^{\circ}=2.5 \sin (180^{\circ}+140^{\circ})

Using the identity $\sin (180^{\circ}+x)=-\sin x$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 \sin 140^{\circ}

Since $\sin 140^{\circ}=\sin (180^{\circ}-40^{\circ})=\sin 40^{\circ}$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 \sin 40^{\circ}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 P

Conclusion

Frequently Asked Questions

Q: What is the purpose of writing trigonometric expressions in terms of a given quantity?

A: The purpose of writing trigonometric expressions in terms of a given quantity is to simplify complex trigonometric expressions and to make them easier to work with. By expressing trigonometric functions in terms of a given quantity, we can eliminate the need to memorize and apply various trigonometric identities.

Q: How do I know which trigonometric identity to use when writing a trigonometric expression in terms of a given quantity?

A: To determine which trigonometric identity to use, you need to analyze the given trigonometric expression and identify the trigonometric function that is being used. You can then use the appropriate trigonometric identity to rewrite the expression in terms of the given quantity.

Q: Can I use any trigonometric identity to write a trigonometric expression in terms of a given quantity?

A: No, not all trigonometric identities can be used to write a trigonometric expression in terms of a given quantity. You need to use the appropriate trigonometric identity that is relevant to the given expression.

Q: How do I simplify a trigonometric expression that is written in terms of a given quantity?

A: To simplify a trigonometric expression that is written in terms of a given quantity, you can use algebraic manipulations and trigonometric identities to rewrite the expression in a simpler form.

Q: Can I use trigonometric expressions written in terms of a given quantity to solve trigonometric equations?

A: Yes, you can use trigonometric expressions written in terms of a given quantity to solve trigonometric equations. By substituting the given quantity into the trigonometric expression, you can solve for the unknown variable.

Q: Are there any limitations to writing trigonometric expressions in terms of a given quantity?

A: Yes, there are limitations to writing trigonometric expressions in terms of a given quantity. For example, you cannot use this technique to solve trigonometric equations that involve complex numbers or irrational numbers.

Q: Can I use trigonometric expressions written in terms of a given quantity to solve problems in physics and engineering?

A: Yes, you can use trigonometric expressions written in terms of a given quantity to solve problems in physics and engineering. By expressing trigonometric functions in terms of a given quantity, you can simplify complex problems and make them easier to solve.

Q: How do I know if a trigonometric expression can be written in terms of a given quantity?

A: To determine if a trigonometric expression can be written in terms of a given quantity, you need to analyze the expression and identify the trigonometric function that is being used. You can then use the appropriate trigonometric identity to rewrite the expression in terms of the given quantity.

Q: Can I use trigonometric expressions written in terms of a given quantity to solve problems in computer science?

A: Yes, you can use trigonometric expressions written in terms of a given quantity to solve problems in computer science. By expressing trigonometric functions in terms of a given quantity, you can simplify complex problems and make them easier to solve.

Example Problems

Problem 1

Express $0.5 \sin 300^{\circ}$ in terms of $P$ .

Solution

We can rewrite $0.5 \sin 300^{\circ}$ as:

0.5 \sin 300^{\circ}=0.5 \sin (180^{\circ}+120^{\circ})

Using the identity $\sin (180^{\circ}+x)=-\sin x$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \sin 120^{\circ}

Since $\sin 120^{\circ}=\sin (180^{\circ}-60^{\circ})=\sin 60^{\circ}$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \sin 60^{\circ}

Since $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-0.5 \frac{\sqrt{3}}{2}

Simplifying this expression, we get:

0.5 \sin 300^{\circ}=-\frac{\sqrt{3}}{4}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

0.5 \sin 300^{\circ}=-\frac{\sqrt{3}}{4} P

Problem 2

Express $2.5 \sin 320^{\circ}$ in terms of $P$ .

Solution

We can rewrite $2.5 \sin 320^{\circ}$ as:

2.5 \sin 320^{\circ}=2.5 \sin (180^{\circ}+140^{\circ})

Using the identity $\sin (180^{\circ}+x)=-\sin x$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 \sin 140^{\circ}

Since $\sin 140^{\circ}=\sin (180^{\circ}-40^{\circ})=\sin 40^{\circ}$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 \sin 40^{\circ}

Since $\sin 40^{\circ}=P$ , we can rewrite this expression as:

2.5 \sin 320^{\circ}=-2.5 P

Conclusion

In this article, we have provided a Q&A section to answer frequently asked questions about writing trigonometric expressions in terms of a given quantity. We have also provided example problems to demonstrate the application of this technique in solving trigonometric equations. By expressing trigonometric functions in terms of a given quantity, we can simplify complex problems and make them easier to solve.