Trigonometric Identities
Overview
The trigonometric functions are tightly bound together by a small set of identities that hold for every angle. The most famous, \(\sin^2 A+\cos^2 A=1\), comes straight from Pythagoras, and two close relatives extend it to the reciprocal functions. These identities are the workhorses for simplifying and proving trigonometric statements.
In this lesson you will meet the three Pythagorean identities and use them to simplify expressions and prove results. Mastering them lets you rewrite a tangled expression in a cleaner form and is essential for solving trigonometric equations later in the course.
Objectives
- Know and use the identities sin^2 A + cos^2 A = 1, sec^2 A = 1 + tan^2 A and cosec^2 A = 1 + cot^2 A.
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Lesson Evaluation
Congratulations on completing the lesson on Trigonometric Identities. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- Which identity is true for every angle A? A. sin A + cos A = 1 B. sin^2 A + cos^2 A = 1 C. sin^2 A - cos^2 A = 1 D. sin A cos A = 1 Answer: B
- The identity sec^2 A equals: A. 1 - tan^2 A B. 1 + tan^2 A C. tan^2 A - 1 D. 1 + cot^2 A Answer: B
- Simplify sec^2 A - tan^2 A. A. 0 B. 1 C. 2 D. tan^2 A Answer: B
- 1 + cot^2 A is equal to: A. sec^2 A B. cosec^2 A C. tan^2 A D. sin^2 A Answer: B
- The expression (sin^2 A + cos^2 A)/cos^2 A simplifies to: A. sin^2 A B. sec^2 A C. cosec^2 A D. 1 Answer: B
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