Simplifying Trigonometric Expressions | Summary and Q&A

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February 16, 2022
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The Organic Chemistry Tutor
Simplifying Trigonometric Expressions

TL;DR

Learn how to simplify trigonometric expressions using reciprocal and pythagorean identities.

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Q: What are reciprocal identities in trigonometry and how can they be used to simplify expressions?

Reciprocal identities, like secant equals 1 over cosine theta and cosecant equals 1 over sine theta, allow us to replace secant or cosecant with fractions that can be simplified further.

Q: How do pythagorean identities help in simplifying trigonometric expressions?

Pythagorean identities, such as sine squared plus cosine squared theta equals one, allow us to replace parts of an expression with their equivalent values, making it easier to simplify the overall expression.

Q: Can you explain how to simplify the expression tangent squared theta plus sine squared theta plus cosine squared theta using pythagorean identities?

By replacing sine squared plus cosine squared with one, the expression simplifies to tangent squared theta plus one. Since one plus tangent squared theta equals secant squared theta, the final simplified expression is secant squared theta.

Q: In the expression cotangent times tan plus cotangent, how can we simplify it further?

By distributing, the expression becomes cotan times tan plus cotangent squared theta. Convert cotangent to cosine over sine and tangent to sine over cosine. The sine and cosine terms will cancel, leaving behind one. Therefore, the expression simplifies to one.

Summary & Key Takeaways

• Trigonometric expressions can be simplified using reciprocal identities such as secant equals 1 over cosine theta and cosecant equals 1 over sine theta.

• Pythagorean identities like sine squared plus cosine squared theta equals one can also be used to simplify expressions.

• By applying these identities, trigonometric expressions can be reduced to simpler forms.