Trigonometric identities

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Understanding these identities is crucial for simplifying trigonometric expressions and solving trigonometric equations.

Types of Trigonometric Identities

There are several important types of trigonometric identities:

• Pythagorean Identities
• Reciprocal Identities
• Quotient Identities
• Co-Function Identities
• Even-Odd Identities

Pythagorean Identities

The Pythagorean identities are derived from the Pythagorean theorem. They relate the squares of the sine and cosine functions:

From this identity, we can derive two other identities:

• 1 + tan^2(θ) = sec^2(θ)
• 1 + cot^2(θ) = csc^2(θ)

Example of Using Pythagorean Identities

To simplify the expression sin^2(θ) + tan^2(θ):

1. Start with the original expression: sin^2(θ) + tan^2(θ).
2. Use the Pythagorean identity for tan^2(θ): tan^2(θ) = sin^2(θ)/cos^2(θ).
3. Substitute into the expression: sin^2(θ) + (sin^2(θ)/cos^2(θ)).
4. Combine the terms: sin^2(θ)(1 + 1/cos^2(θ)).
5. Express in terms of cosine: sin^2(θ)(cos^2(θ) + 1)/cos^2(θ).
6. Thus, the simplified form is: sin^2(θ)(sec^2(θ)).

Reciprocal Identities

Reciprocal identities express trigonometric functions in terms of their reciprocals:

• sin(θ) = 1/csc(θ)
• cos(θ) = 1/sec(θ)
• tan(θ) = 1/cot(θ)

Example of Using Reciprocal Identities

To express sin(θ) in terms of csc(θ):

1. Use the reciprocal identity: sin(θ) = 1/csc(θ).
2. This means that if csc(θ) = 2, then sin(θ) = 1/2.

Quotient Identities

Quotient identities relate sine and cosine to tangent and cotangent:

• tan(θ) = sin(θ)/cos(θ)
• cot(θ) = cos(θ)/sin(θ)

Example of Using Quotient Identities

To express tan(θ) in terms of sin(θ) and cos(θ):

1. Start with the definition: tan(θ) = sin(θ)/cos(θ).
2. If sin(θ) = 3/5 and cos(θ) = 4/5, then:
3. tan(θ) = (3/5)/(4/5) = 3/4.

Co-Function Identities

Co-function identities relate the trigonometric functions of complementary angles:

• sin(90° - θ) = cos(θ)
• cos(90° - θ) = sin(θ)
• tan(90° - θ) = cot(θ)
• csc(90° - θ) = sec(θ)
• sec(90° - θ) = csc(θ)
• cot(90° - θ) = tan(θ)

Example of Using Co-Function Identities

To find sin(30°) using the co-function identity:

1. Use the identity: sin(90° - θ) = cos(θ).
2. Thus, sin(30°) = cos(60°).
3. Since cos(60°) = 1/2, sin(30°) = 1/2.

Even-Odd Identities

Even-odd identities describe how trigonometric functions behave with negative angles:

• sin(-θ) = -sin(θ) (sine is an odd function)
• cos(-θ) = cos(θ) (cosine is an even function)
• tan(-θ) = -tan(θ) (tangent is an odd function)

Example of Using Even-Odd Identities

To evaluate sin(-45°):

1. Use the identity: sin(-θ) = -sin(θ).
2. Thus, sin(-45°) = -sin(45°).
3. Since sin(45°) = √2/2, sin(-45°) = -√2/2.

Combining Identities

Often, you will need to combine different identities to simplify expressions or solve equations. For example, to simplify the expression:

sin^2(θ) + cos^2(θ) + tan^2(θ)

1. Use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
2. Then, replace tan^2(θ) with sin^2(θ)/cos^2(θ).
3. The expression becomes 1 + sin^2(θ)/cos^2(θ).

Summary

• Trigonometric identities are essential for simplifying expressions and solving equations.
• Types include Pythagorean, reciprocal, quotient, co-function, and even-odd identities.
• Combining identities can help in simplifying complex expressions.

Check your understanding

1. Prove the identity: 1 + tan^2(θ) = sec^2(θ).
2. Using the co-function identity, find cos(30°).
3. Simplify the expression: sin^2(θ) + cos^2(θ) + 1.
4. Evaluate tan(-30°) using the even-odd identity.