Choose the appropriate identity

Oct 13, 2025 by magicalceylontours.com 84 views

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. One of the key concepts in trigonometry is the use of compound angle formulas to simplify and solve trigonometric expressions. In this article, we will prove two trigonometric identities using compound angle formulas: $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$ and $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$ .

Compound Angle Formulas

Before we proceed to prove the given identities, let's first recall the compound angle formulas for sine and cosine functions.

Sine Function

The compound angle formula for the sine function is given by:

\sin (A+B) = \sin A \cos B + \cos A \sin B

Cosine Function

The compound angle formula for the cosine function is given by:

\cos (A+B) = \cos A \cos B - \sin A \sin B

Proof of Identity a) $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$

To prove the identity $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$ , we will use the compound angle formula for the sine function.

We know that $\sin 270^{\circ} = -1$ and $\cos 270^{\circ} = 0$ . Using the compound angle formula, we can write:

\sin \left(270^{\circ}+\theta\right) = \sin 270^{\circ} \cos \theta + \cos 270^{\circ} \sin \theta

Substituting the values of $\sin 270^{\circ}$ and $\cos 270^{\circ}$ , we get:

\sin \left(270^{\circ}+\theta\right) = (-1) \cos \theta + (0) \sin \theta

Simplifying the expression, we get:

\sin \left(270^{\circ}+\theta\right) = -\cos \theta

Therefore, we have proved the identity $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$ .

Proof of Identity b) $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$

To prove the identity $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$ , we will use the compound angle formula for the cosine function.

We know that $\cos 180^{\circ} = -1$ and $\sin 180^{\circ} = 0$ . Using the compound angle formula, we can write:

\cos \left(180^{\circ}+\theta\right) = \cos 180^{\circ} \cos \theta - \sin 180^{\circ} \sin \theta

Substituting the values of $\cos 180^{\circ}$ and $\sin 180^{\circ}$ , we get:

\cos \left(180^{\circ}+\theta\right) = (-1) \cos \theta - (0) \sin \theta

Simplifying the expression, we get:

\cos \left(180^{\circ}+\theta\right) = -\cos \theta

Therefore, we have proved the identity $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$ .

Conclusion

In this article, we have proved two trigonometric identities using compound angle formulas: $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$ and $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$ . These identities are essential in trigonometry and have numerous applications in various fields. We hope that this article has provided a clear understanding of the compound angle formulas and their applications in proving trigonometric identities.

References

[1] Kern, W. F., & Bland, J. R. (1970). Solid analytic trigonometry. Houghton Mifflin.
[2] Larson, R. E., & Hostetler, R. P. (1983). Trigonometry. Houghton Mifflin.
[3] Swokowski, E. W. (1987). Trigonometry. Houghton Mifflin.

Glossary

Compound Angle Formula: A formula that expresses the sine or cosine of the sum of two angles in terms of the sines and cosines of the individual angles.
Trigonometric Identity: An equation that is true for all values of the variables involved.
Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.
Frequently Asked Questions (FAQs) on Trigonometric Identities ================================================================

Introduction

In our previous article, we proved two trigonometric identities using compound angle formulas: $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$ and $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$ . In this article, we will address some of the frequently asked questions (FAQs) related to these identities.

Q&A

Q: What is the difference between a trigonometric identity and a trigonometric equation?

A: A trigonometric identity is an equation that is true for all values of the variables involved, whereas a trigonometric equation is an equation that is not necessarily true for all values of the variables involved.

Q: How do I know if a trigonometric identity is true or not?

A: To determine if a trigonometric identity is true or not, you can use various methods such as:

Substitution: Substitute specific values of the variables into the identity and check if the equation holds true.
Graphical analysis: Plot the graphs of the functions involved in the identity and check if they intersect at the same point.
Algebraic manipulation: Manipulate the identity algebraically to simplify it and check if it reduces to a known identity.

Q: Can I use trigonometric identities to solve trigonometric equations?

A: Yes, you can use trigonometric identities to solve trigonometric equations. By applying the appropriate identity, you can simplify the equation and solve for the unknown variable.

Q: How do I apply trigonometric identities to solve problems?

A: To apply trigonometric identities to solve problems, follow these steps:

Identify the problem: Read the problem carefully and identify the trigonometric functions involved.
Choose the appropriate identity: Select the trigonometric identity that is relevant to the problem.
Apply the identity: Substitute the values of the variables into the identity and simplify the equation.
Solve for the unknown variable: Solve for the unknown variable using the simplified equation.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

Pythagorean identities: $\sin^2 \theta + \cos^2 \theta = 1$ and $\tan^2 \theta + 1 = \sec^2 \theta$
Sum and difference identities: $\sin (A+B) = \sin A \cos B + \cos A \sin B$ and $\cos (A+B) = \cos A \cos B - \sin A \sin B$
Double angle identities: $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos 2\theta = 1 - 2\sin^2 \theta$

Q: How do I prove trigonometric identities?

A: To prove trigonometric identities, you can use various methods such as:

Algebraic manipulation: Manipulate the identity algebraically to simplify it and check if it reduces to a known identity.
Graphical analysis: Plot the graphs of the functions involved in the identity and check if they intersect at the same point.
Substitution: Substitute specific values of the variables into the identity and check if the equation holds true.

Conclusion

In this article, we have addressed some of the frequently asked questions (FAQs) related to trigonometric identities. We hope that this article has provided a clear understanding of the concepts and methods involved in proving and applying trigonometric identities.

References

[1] Kern, W. F., & Bland, J. R. (1970). Solid analytic trigonometry. Houghton Mifflin.
[2] Larson, R. E., & Hostetler, R. P. (1983). Trigonometry. Houghton Mifflin.
[3] Swokowski, E. W. (1987). Trigonometry. Houghton Mifflin.

Glossary

Trigonometric identity: An equation that is true for all values of the variables involved.
Trigonometric equation: An equation that is not necessarily true for all values of the variables involved.
Compound angle formula: A formula that expresses the sine or cosine of the sum of two angles in terms of the sines and cosines of the individual angles.