Evaluate { \cos \left(90^ \circ}-\theta\right)$}$.Answer Options A. { \csc \theta$ $B. { \tan \theta$}$C. { \sec \theta$}$D. { \cos \theta$}$E. { \cot \theta$}$F. { \sin \theta$}$

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Understanding the Problem

When evaluating trigonometric functions, it's essential to recall the fundamental identities and relationships between different trigonometric ratios. In this problem, we are asked to evaluate the cosine of an expression involving a subtraction of an angle from $90^{\circ}$. This is a classic example of using trigonometric identities to simplify and evaluate expressions.

Recalling Trigonometric Identities

To tackle this problem, we need to recall the trigonometric identity for the cosine of a difference of angles. This identity states that $\cos (A - B) = \cos A \cos B + \sin A \sin B$. However, in this case, we can use a more straightforward approach by utilizing the fact that $\cos (90^{\circ} - \theta) = \sin \theta$.

Applying the Identity

Using the identity mentioned above, we can rewrite the expression as $\cos (90^{\circ} - \theta) = \sin \theta$. This is because the cosine of an angle that is $90^{\circ}$ minus another angle is equal to the sine of the latter angle.

Evaluating the Expression

Now that we have simplified the expression to $\sin \theta$, we can evaluate it. The sine of an angle is a fundamental trigonometric ratio that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.

Conclusion

In conclusion, the correct answer to the problem is $\boxed{\sin \theta}$. This is because we used the trigonometric identity for the cosine of a difference of angles to simplify the expression and arrived at the sine of the angle.

Answer Options

For the sake of completeness, let's review the answer options:

A. $\csc \theta$ B. $\tan \theta$ C. $\sec \theta$ D. $\cos \theta$ E. $\cot \theta$ F. $\sin \theta$

As we have seen, the correct answer is F. $\sin \theta$.

Additional Examples and Practice

To further reinforce your understanding of trigonometric identities and their applications, try the following examples:

• Evaluate $\cos (180^{\circ} - \theta)$.
• Simplify $\sin (90^{\circ} + \theta)$.
• Evaluate $\tan (180^{\circ} - \theta)$.

These examples will help you develop a deeper understanding of trigonometric identities and their applications in various mathematical contexts.

Final Thoughts

In this article, we evaluated the expression $\cos (90^{\circ} - \theta)$ using trigonometric identities and arrived at the correct answer, $\sin \theta$. This problem serves as a reminder of the importance of recalling and applying fundamental trigonometric identities to simplify and evaluate expressions. By practicing and mastering these identities, you will become more confident and proficient in solving trigonometric problems.

Introduction

Trigonometric identities are the building blocks of trigonometry, and mastering them is essential for solving problems in mathematics, physics, engineering, and other fields. In this article, we will provide a comprehensive Q&A guide to help you understand and apply trigonometric identities with confidence.

Q1: What is a trigonometric identity?

A1: A trigonometric identity is a mathematical statement that expresses the relationship between different trigonometric ratios, such as sine, cosine, and tangent, for a given angle.

Q2: What are the basic trigonometric identities?

A2: The basic trigonometric identities include:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$

Q3: How do I simplify a trigonometric expression using identities?

A3: To simplify a trigonometric expression using identities, follow these steps:

1. Identify the trigonometric ratios involved in the expression.
2. Look for identities that relate these ratios.
3. Apply the identities to simplify the expression.

Q4: What is the Pythagorean identity?

A4: The Pythagorean identity is $\sin^2 \theta + \cos^2 \theta = 1$. This identity is used to express the relationship between sine and cosine.

Q5: How do I evaluate $\cos (A - B)$?

A5: To evaluate $\cos (A - B)$, use the identity $\cos (A - B) = \cos A \cos B + \sin A \sin B$.

Q6: What is the sum and difference formula for sine and cosine?

A6: The sum and difference formula for sine and cosine are:

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

Q7: How do I evaluate $\tan (A + B)$?

A7: To evaluate $\tan (A + B)$, use the identity $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$.

Q8: What is the double-angle formula for sine and cosine?

A8: The double-angle formula for sine and cosine are:

• $\sin 2A = 2 \sin A \cos A$
• $\cos 2A = \cos^2 A - \sin^2 A$

Q9: How do I evaluate $\sin (A - B)$?

A9: To evaluate $\sin (A - B)$, use the identity $\sin (A - B) = \sin A \cos B - \cos A \sin B$.

Q10: What is the triple-angle formula for sine and cosine?

A10: The triple-angle formula for sine and cosine are:

• $\sin 3A = 3 \sin A - 4 \sin^3 A$
• $\cos 3A = 4 \cos^3 A - 3 \cos A$

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand and apply trigonometric identities with confidence. By mastering these identities, you will become more proficient in solving trigonometric problems and will be better equipped to tackle complex mathematical challenges.

Additional Resources

For further practice and review, try the following resources:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram Alpha: Trigonometry

Final Thoughts

Trigonometric identities are the foundation of trigonometry, and mastering them is essential for success in mathematics, physics, engineering, and other fields. By practicing and applying these identities, you will become more confident and proficient in solving trigonometric problems.