Pythagorean Identity

Grade 11 · trigonometry · 84 practice problems · read aloud

🔊 Listen to this explanation

The Pythagorean Identity: sin²θ + cos²θ = 1

🔍 What is it and why is it useful?

The Pythagorean Identity is a fundamental relationship between sine and cosine for any angle θ. It's derived directly from the Pythagorean Theorem applied to the unit circle. This identity is incredibly useful for:

  • Simplifying complex trigonometric expressions
  • Finding the value of one trigonometric function when you know another
  • Proving other trigonometric identities
  • Solving trigonometric equations

📝 Step-by-Step Problem Solving

  1. Identify what you know: Determine which trigonometric function's value is given.
  2. Choose the correct form: Use sin²θ + cos²θ = 1, or rearrange it to sin²θ = 1 - cos²θ or cos²θ = 1 - sin²θ.
  3. Substitute the known value: Plug in the given number.
  4. Solve for the unknown: Remember to consider the sign (±) based on the quadrant.
  5. Simplify your answer: Rationalize denominators if necessary.

✨ Worked Examples

Example 1: If sinθ = 3/5 and θ is in Quadrant II, find cosθ.

Step 1: Start with the identity: sin²θ + cos²θ = 1

Step 2: Substitute: (3/5)² + cos²θ = 1 → 9/25 + cos²θ = 1

Step 3: Solve for cos²θ: cos²θ = 1 - 9/25 = 16/25

Step 4: Take square root: cosθ = ±4/5. Since θ is in Quadrant II (where cosine is negative), the answer is cosθ = -4/5.

Example 2: Simplify the expression: 1 - cos²θ

From the identity sin²θ + cos²θ = 1, we can rearrange to get 1 - cos²θ = sin²θ.

Therefore, the expression simplifies directly to sin²θ.

⚠️ Common Mistakes to Avoid

Forgetting the ± sign: When you take a square root, you get both positive and negative solutions. Always check the quadrant to determine the correct sign.

Misplacing the exponent: sin²θ means (sinθ)². It's not sin(θ²)!

Assuming all angles are in Quadrant I: The quadrant determines the sign of sine, cosine, and tangent. Don't default to positive!

💡 Tips & Tricks

Memory Aid: Think "SOH-CAH-TOA" on the unit circle. The hypotenuse is 1, so (Opposite)² + (Adjacent)² = 1² becomes sin²θ + cos²θ = 1.

Quick Check: If sinθ and cosθ are both between -1 and 1, their squares are positive and sum to 1. This is a good sanity check for your answers.

Derived Identities: Remember you can also divide the entire identity by sin²θ or cos²θ to get 1 + cot²θ = csc²θ and tan²θ + 1 = sec²θ.

🎯 How to Practice

  • Start with basic problems: Given sinθ, find cosθ (and vice versa) for different quadrants.
  • Practice simplifying expressions like (1 - sin²θ) / cosθ.
  • Combine the identity with other trig concepts, like verifying identities or solving equations.
  • Use online resources or textbook exercises that provide immediate feedback.

Practice problems

6 of the 84, worked through step by step — try them before opening the answer.

1 sin²θ + cos²θ = ?

Hint: This fundamental trigonometric identity holds true for any angle value and relates the squares of the sine and cosine functions.

Show the answer

Answer: 1

  1. Recall the Pythagorean identity from trigonometry. This identity states that for any angle θ, the square of the sine of θ plus the square of the cosine of θ equals 1.
  2. Write the identity in equation form. sin²θ + cos²θ = 1
  3. Explanation. This identity holds true for all values of θ. It comes from the definition of sine and cosine on the unit circle, where the hypotenuse is 1, so by the Pythagorean theorem: (opposite side)² + (adjacent side)² = (hypotenuse)² But opposite side = sin θ, adjacent side = cos θ, hypotenuse = 1. So (sin θ)² + (cos θ)² = 1², which is sin²θ + cos²θ = 1.
  4. Conclusion. Therefore,

We are given the problem: sin²θ + cos²θ = ? the answer is always 1, regardless of the value of θ. Final answer: 1

2 Given sin θ = 4/5, find cos θ using sin²θ + cos²θ = 1

Hint: Use the Pythagorean identity to relate sine and cosine, then solve for the unknown trigonometric function.

Show the answer

Answer: 3/5

  1. Start with the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute the given value: (4/5)² + cos²θ = 1
  3. Calculate (4/5)² = 16/25
  4. Write the equation: 16/25 + cos²θ = 1
  5. Subtract 16/25 from both sides: cos²θ = 1 - 16/25
  6. Convert 1 to 25/25: cos²θ = 25/25 - 16/25
  7. Simplify: cos²θ = 9/25
  8. Take the square root of both sides: cos θ = ±3/5
  9. Since the problem doesn't specify the quadrant, we take the positive value: cos θ = 3/5 Final answer: 3/5

3 Given sin θ = 6/10, find cos θ using sin²θ + cos²θ = 1

Hint: Substitute the given sine value into the identity and solve for cosine, remembering that cosine could be positive or negative depending on the quadrant.

Show the answer

Answer: 8/10

  1. Start with the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute the given value sin θ = 6/10: (6/10)² + cos²θ = 1
  3. Calculate (6/10)² = 36/100
  4. Write the equation: 36/100 + cos²θ = 1
  5. Subtract 36/100 from both sides: cos²θ = 1 - 36/100 = 100/100 - 36/100 = 64/100
  6. Take the square root of both sides: cos θ = ±√(64/100) = ±8/10
  7. Since the problem doesn't specify the quadrant, we typically take the positive value: cos θ = 8/10 Final answer: 8/10

4 Given sin θ = 9/41, find cos θ using sin²θ + cos²θ = 1

Hint: Use the Pythagorean identity to relate sine and cosine, then solve for the unknown trigonometric function.

Show the answer

Answer: 40/41

  1. Start with the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute the given value: (9/41)² + cos²θ = 1
  3. Calculate (9/41)² = 81/1681
  4. Write the equation: 81/1681 + cos²θ = 1
  5. Subtract 81/1681 from both sides: cos²θ = 1 - 81/1681
  6. Convert 1 to 1681/1681: cos²θ = 1681/1681 - 81/1681
  7. Subtract: cos²θ = 1600/1681
  8. Take the square root of both sides: cos θ = ±√(1600/1681)
  9. Simplify: cos θ = ±40/41
  10. Since the problem doesn't specify a quadrant, we take the positive value as the principal answer: cos θ = 40/41

5 Given cos θ = 8/17, find sin θ using sin²θ + cos²θ = 1

Hint: Remember that the Pythagorean identity relates the squares of sine and cosine. Substitute the given value and solve for the unknown trigonometric function.

Show the answer

Answer: 15/17

  1. Start with the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute the given value cos θ = 8/17 into the equation: sin²θ + (8/17)² = 1
  3. Calculate (8/17)² = 64/289
  4. The equation becomes: sin²θ + 64/289 = 1
  5. Subtract 64/289 from both sides: sin²θ = 1 - 64/289
  6. Convert 1 to 289/289: sin²θ = 289/289 - 64/289
  7. Simplify: sin²θ = 225/289
  8. Take the square root of both sides: sin θ = ±√(225/289)
  9. Simplify: sin θ = ±15/17
  10. Since no quadrant is specified, we take the positive value: sin θ = 15/17 Final answer: 15/17

6 Given sin θ = 5/13, find cos θ using sin²θ + cos²θ = 1

Hint: Use the Pythagorean identity to find the missing trigonometric function value. Remember to consider the sign based on the quadrant.

Show the answer

Answer: 12/13

  1. Start with the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute the given value: (5/13)² + cos²θ = 1
  3. Calculate (5/13)² = 25/169
  4. Write the equation: 25/169 + cos²θ = 1
  5. Subtract 25/169 from both sides: cos²θ = 1 - 25/169
  6. Calculate 1 - 25/169 = 169/169 - 25/169 = 144/169
  7. Take the square root: cos θ = ±√(144/169) = ±12/13
  8. Since the problem doesn't specify a quadrant, we take the positive value: cos θ = 12/13 Final answer: 12/13
Practise this topic — 10 free problems, no signup →