Subjects

Algebra 2

Trigonometry

The Pythagorean Identity

Algebra 2

Dec 10, 2025

199

5 pages

Mastering Trigonometric Identities for Algebra 2

Layla Guthrie @laylaguthrie_aoly

Trigonometric identities form the backbone of advanced math and physics. These mathematical relationships between trigonometric functions remain true... Show more

14-1 Trignometric Identities
sin → Cosecant (csc) +
secant (sec) H
>Cotangent (cot)
-COS
A tan
->
Trig Identity: a trig equation that is tru

Understanding Trigonometric Identities

Trigonometric identities are equations involving trig functions that are true for all values where the expressions are defined. The reciprocal identities establish relationships between primary and secondary trig functions

sin θ = 1/csc θ (and vice versa)
cos θ = 1/sec θ (and vice versa)
tan θ = 1/cot θ (and vice versa)

The tangent identity $tan θ = sin θ/cos θ$ and cotangent identity $cot θ = cos θ/sin θ$ connect these functions. When verifying identities, work on the more complex side while leaving the simpler side alone.

The Pythagorean identities are three fundamental relationships

cos²θ + sin²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

💡 Remember that sin²θ means (sin θ)² - the entire function is squared, not just the angle!

These identities can be rearranged to isolate different functions. For instance, sin²θ = 1 - cos²θ helps when you need to replace sine with cosine expressions.

Verifying Trig Identities

Verifying trig identities requires careful algebraic manipulation. Start by working with the more complex side of the equation until it matches the simpler side. A common strategy is to convert all expressions into the same functions (all sines and cosines).

When working with fractions containing trig functions, look for opportunities to multiply by helpful forms of 1 $like sin θ/sin θ$ . This helps create common denominators or simplify the expression.

For example, to verify csc θ - sin θ = cot θ·cos θ

Start with the left side csc θ - sin θ
Rewrite using reciprocal identity 1/sin θ - sin θ
Find a common denominator $1 - sin²θ$ /sin θ
Apply Pythagorean identity $1 - sin²θ = cos²θ$ cos²θ/sin θ
Rewrite as cos θ· $cos θ/sin θ$ = cos θ·cot θ

🔑 The key to verification is patience! Always look for opportunities to apply the fundamental identities to transform expressions.

Solving Trig Equations

Trig equations allow you to find angle values that satisfy specific conditions. When given one trig value and a quadrant, you can find other trig values using identities.

For example, if sin θ = 0.2 and tan θ > 0, you can find cos θ using the Pythagorean identity

Start with sin²θ + cos²θ = 1
Substitute the known value (0.2)² + cos²θ = 1
Solve cos²θ = 0.96
Take the square root cos θ = ±0.98

Since tan θ = sin θ/cos θ > 0, and sin θ = 0.2 (positive), cos θ must also be positive. Therefore, cos θ = 0.98.

When verifying identities like tan θ·cot θ = 1, substitute the definitions $sin θ/cos θ$ · $cos θ/sin θ$ = 1 ✓

💡 When solving trig equations, always consider the domain restrictions and quadrant information to determine the correct solutions!

Advanced Identities and Applications

Mastering trig identities opens doors to solving more complex problems. When dealing with multiple identities, use strategic substitutions to simplify expressions.

For example, to evaluate cot θ·tan²θ

Substitute definitions $cos θ/sin θ$ · $sin θ/cos θ$ ²
Simplify $cos θ/sin θ$ · $sin²θ/cos²θ$ = sin θ/cos θ = tan θ

Trig equations like 4sin²x - 1 = 0 can be solved systematically

Rearrange sin²x = 1/4
Take the square root sin x = ±1/2
Find angles x = 30°, 150°, 210°, 330°

When applying identities to real problems, remember that angles can be expressed in degrees or radians. For instance, 285° = 19π/12 radians.

🧠 Trigonometric functions are periodic, which means their values repeat every 360° (or 2π radians). This is why most trig equations have multiple solutions!

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Subjects

Algebra 2

Trigonometry

The Pythagorean Identity

Algebra 2

•

199

•

Dec 10, 2025

•

5 pages

Mastering Trigonometric Identities for Algebra 2

Layla Guthrie

@laylaguthrie_aoly

Trigonometric identities form the backbone of advanced math and physics. These mathematical relationships between trigonometric functions remain true for any angle value and serve as powerful tools for simplifying complex expressions and solving equations.

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Understanding Trigonometric Identities

sin θ = 1/csc θ (and vice versa)
cos θ = 1/sec θ (and vice versa)
tan θ = 1/cot θ (and vice versa)

The Pythagorean identities are three fundamental relationships:

cos²θ + sin²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

💡 Remember that sin²θ means (sin θ)² - the entire function is squared, not just the angle!

These identities can be rearranged to isolate different functions. For instance, sin²θ = 1 - cos²θ helps when you need to replace sine with cosine expressions.

Sign up to see the contentIt's free!

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Verifying Trig Identities

For example, to verify csc θ - sin θ = cot θ·cos θ:

Start with the left side: csc θ - sin θ
Rewrite using reciprocal identity: 1/sin θ - sin θ
Find a common denominator: $1 - sin²θ$ /sin θ
Apply Pythagorean identity $1 - sin²θ = cos²θ$ : cos²θ/sin θ
Rewrite as cos θ· $cos θ/sin θ$ = cos θ·cot θ

🔑 The key to verification is patience! Always look for opportunities to apply the fundamental identities to transform expressions.

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Solving Trig Equations

Trig equations allow you to find angle values that satisfy specific conditions. When given one trig value and a quadrant, you can find other trig values using identities.

For example, if sin θ = 0.2 and tan θ > 0, you can find cos θ using the Pythagorean identity:

Start with sin²θ + cos²θ = 1
Substitute the known value: (0.2)² + cos²θ = 1
Solve: cos²θ = 0.96
Take the square root: cos θ = ±0.98

Since tan θ = sin θ/cos θ > 0, and sin θ = 0.2 (positive), cos θ must also be positive. Therefore, cos θ = 0.98.

When verifying identities like tan θ·cot θ = 1, substitute the definitions: $sin θ/cos θ$ · $cos θ/sin θ$ = 1 ✓

💡 When solving trig equations, always consider the domain restrictions and quadrant information to determine the correct solutions!

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

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Advanced Identities and Applications

Mastering trig identities opens doors to solving more complex problems. When dealing with multiple identities, use strategic substitutions to simplify expressions.

For example, to evaluate cot θ·tan²θ:

Substitute definitions: $cos θ/sin θ$ · $sin θ/cos θ$ ²
Simplify: $cos θ/sin θ$ · $sin²θ/cos²θ$ = sin θ/cos θ = tan θ

Trig equations like 4sin²x - 1 = 0 can be solved systematically:

Rearrange: sin²x = 1/4
Take the square root: sin x = ±1/2
Find angles: x = 30°, 150°, 210°, 330°

When applying identities to real problems, remember that angles can be expressed in degrees or radians. For instance, 285° = 19π/12 radians.

🧠 Trigonometric functions are periodic, which means their values repeat every 360° (or 2π radians). This is why most trig equations have multiple solutions!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

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