Pythagorean Identity Applications

Grade 11 · trigonometry · 84 practice problems · read aloud

🔊 Listen to this explanation

🧮 What is the Pythagorean Identity?

The Pythagorean Identity is a fundamental relationship between the sine and cosine of an angle. It states that for any angle θ:

sin²θ + cos²θ = 1

This identity is incredibly useful because it allows you to find one trigonometric function value if you know the other, and it's essential for simplifying complex trigonometric expressions and solving equations.

📝 Step-by-Step Problem Solving Guide

  1. Identify what you know: Determine which trig function value is given and which quadrant the angle is in (this affects the sign!).
  2. Plug into the identity: Substitute the known value into sin²θ + cos²θ = 1.
  3. Solve for the unknown: Rearrange the equation algebraically to find the missing value.
  4. Determine the sign: Use the quadrant information to choose the correct sign (+ or -) for your answer.

🔍 Worked Examples

Example 1: Finding Cosine from Sine

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

Step 1: We know sinθ = 3/5. In QII, cosine is negative.

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

Step 3: cos²θ = 1 - 9/25 = 16/25 → cosθ = ±4/5

Step 4: Since θ is in QII, cosθ = -4/5

Example 2: Simplifying an Expression

Problem: Simplify the expression: 1 - cos²θ

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

sin²θ = 1 - cos²θ

Therefore, 1 - cos²θ simplifies directly to sin²θ.

⚠️ Common Mistakes to Avoid

  • Forgetting the ± sign: Always consider the algebraic solution first (both + and -), THEN use the quadrant to pick the correct one.
  • Ignoring the quadrant: The quadrant is crucial! In QII: sin+, cos-; QIII: sin-, cos-; QIV: sin-, cos+.
  • Misplacing the square: Remember the identity is sin²θ + cos²θ = 1, not sinθ + cosθ = 1.

💡 Tips & Tricks

  • Memory Aid: Think "SOH CAH TOA" but for the identity: "S² + C² = 1".
  • Quick Check: Your final sine and cosine values must satisfy the identity. If sin²θ + cos²θ doesn't equal 1, you made a mistake.
  • Derived Identities: Remember you can also write the identity as sin²θ = 1 - cos²θ or cos²θ = 1 - sin²θ.

🎯 How to Practice

To master this skill, try these practice strategies:

  • Start with basic "find the missing value" problems, making sure to vary the quadrant.
  • Practice simplifying expressions like (1 - sin²θ) / cosθ.
  • Combine it with other skills, like proving trigonometric identities.
  • Create flashcards with different scenarios (e.g., "sinθ = -1/3, QIII, find cosθ").

Practice problems

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

1 If sin θ = 3/5 and θ is in quadrant II, find cos θ = ?

Hint: Use the Pythagorean identity that relates sine and cosine, and consider the sign based on the quadrant location.

Show the answer

Answer: -4/5

  1. Recall the Pythagorean identity. The identity is: sin² θ + cos² θ = 1.
  2. Substitute the given value of sin θ into the identity. We are told sin θ = 3/5. So: (3/5)² + cos² θ = 1.
  3. Calculate (3/5)². (3/5)² = 9/25. So the equation becomes: 9/25 + cos² θ = 1.
  4. Solve for cos² θ. Subtract 9/25 from both sides: cos² θ = 1 - 9/25. Write 1 as 25/25: cos² θ = 25/25 - 9/25 = 16/25.
  5. Take the square root to find cos θ. cos θ = ±√(16/25) = ±(4/5).
  6. Determine the correct sign using the quadrant information. θ is in quadrant II. In quadrant II, sine is positive and cosine is negative. Since sin θ = 3/5 is positive, that matches quadrant II. Therefore, cos θ must be negative.
  7. Final answer. cos θ = -4/5.

2 If sin θ = -4/5 and θ is in quadrant IV, find cos θ = ?

Hint: Use the Pythagorean identity and consider the sign based on the quadrant

Show the answer

Answer: 3/5

  1. Use the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute sin θ = -4/5: (-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. Calculate 1 - 16/25 = 25/25 - 16/25 = 9/25
  7. Take square root: cos θ = ±√(9/25) = ±3/5
  8. Since θ is in quadrant IV, cosine is positive, so cos θ = 3/5

The answer is 3/5.

3 If sin θ = -3/5 and θ is in quadrant IV, find cos θ = ?

Hint: Consider the Pythagorean identity and the sign conventions for trigonometric functions in different quadrants.

Show the answer

Answer: 4/5

  1. Use the Pythagorean identity: sin²θ + cos²θ = 1
  2. Substitute sin θ = -3/5: (-3/5)² + cos²θ = 1
  3. Calculate (-3/5)² = 9/25: 9/25 + cos²θ = 1
  4. Subtract 9/25 from both sides: cos²θ = 1 - 9/25 = 16/25
  5. Take the square root: cos θ = ±√(16/25) = ±4/5
  6. Determine the sign: In quadrant IV, cosine is positive, so cos θ = 4/5

The answer is 4/5.

4 If tan θ = 3/4 and θ is in quadrant III, find sin θ = ?

Hint: In quadrant III, both sine and cosine are negative. Use the relationship between tangent, sine, and cosine along with the Pythagorean identity.

Show the answer

Answer: -3/5

  1. Given tan θ = 3/4, we know that sin θ / cos θ = 3/4
  2. Let sin θ = 3k and cos θ = 4k for some constant k
  3. Apply the Pythagorean identity: sin²θ + cos²θ = 1
  4. Substitute: (3k)² + (4k)² = 1
  5. Calculate: 9k² + 16k² = 1
  6. Combine: 25k² = 1
  7. Solve for k²: k² = 1/25
  8. Solve for k: k = ±1/5
  9. Since θ is in quadrant III, both sin θ and cos θ are negative, so k must be negative: k = -1/5
  10. Find sin θ = 3k = 3(-1/5) = -3/5

The answer is -3/5.

5 If sin(θ) = 3/5 and θ is in quadrant II, find cos(θ) = ?

Hint: Use the Pythagorean identity that relates sine and cosine. Consider the sign of the trigonometric function based on the quadrant.

Show the answer

Answer: -4/5

  1. Recall the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
  2. Substitute sin(θ) = 3/5 into the identity: (3/5)^2 + cos^2(θ) = 1.
  3. Calculate (3/5)^2: (3/5)^2 = 9/25.
  4. Write the equation: 9/25 + cos^2(θ) = 1.
  5. Subtract 9/25 from both sides: cos^2(θ) = 1 - 9/25.
  6. Write 1 as 25/25: cos^2(θ) = 25/25 - 9/25 = 16/25.
  7. Take the square root of both sides: cos(θ) = ±√(16/25) = ±(4/5).
  8. Determine the correct sign using the quadrant information. θ is in quadrant II. In quadrant II, cosine is negative and sine is positive. Since sin(θ) is positive (3/5), that matches quadrant II. Therefore, cos(θ) must be negative.
  9. Choose the negative sign: cos(θ) = -4/5. Final answer: -4/5

We are given: sin(θ) = 3/5 and θ is in quadrant II.

6 If cos θ = -4/5 and θ is in quadrant III, find sin θ = ?

Hint: Use the Pythagorean identity and consider the sign based on the quadrant

Show the answer

Answer: -3/5

  1. Use the Pythagorean identity sin²θ + cos²θ = 1
  2. Substitute cos θ = -4/5 into the identity: sin²θ + (-4/5)² = 1
  3. Calculate (-4/5)² = 16/25
  4. sin²θ + 16/25 = 1
  5. Subtract 16/25 from both sides: sin²θ = 1 - 16/25 = 9/25
  6. Take the square root: sin θ = ±3/5
  7. Since θ is in quadrant III where sine is negative, sin θ = -3/5

The answer is -3/5.

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