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Pythagorean Identities

A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.

The three Pythagorean identities come from sin^2(theta)+cos^2(theta)=1: dividing through by cos^2 gives 1+tan^2(theta)=sec^2(theta), and dividing by sin^2 gives 1+cot^2(theta)=csc^2(theta). You use them to simplify expressions like 1-sin^2(theta) and, given one ratio plus a quadrant, to solve for another ratio while picking the correct sign.

What you'll learn

Why it matters: When an oscillation or wave is described by one component (say its sine), the Pythagorean identity recovers the other component, so engineers and physicists can convert between sine and cosine forms without remeasuring.

Worked example

Problem. If sin(theta)=3/5 and theta is in Quadrant II, find cos(theta).

  1. Use sin^2(theta)+cos^2(theta)=1, so cos^2(theta)=1-9/25=16/25.
  2. Take the square root: cos(theta)=plus or minus 4/5.
  3. In Quadrant II cosine is negative, so cos(theta)=-4/5.

Answer: -4/5

Practice problems

1. Simplify 1 - sin^2(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify 1 - sin^2(theta).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Start from sin^2(theta)+cos^2(theta)=1.
  4. Subtract sin^2(theta) from both sides.
  5. 1 - sin^2(theta) = cos^2(theta).
  6. Check the result by substituting or estimating: the response should match cos^2(theta) and make sense in the original problem.

Answer: cos^2(theta)

2. Simplify 1 - cos^2(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify 1 - cos^2(theta).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Start from sin^2(theta)+cos^2(theta)=1.
  4. Subtract cos^2(theta) from both sides.
  5. 1 - cos^2(theta) = sin^2(theta).
  6. Check the result by substituting or estimating: the response should match sin^2(theta) and make sense in the original problem.

Answer: sin^2(theta)

3. Write the Pythagorean identity that equals sec^2(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write the Pythagorean identity that equals sec^2(theta).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide sin^2+cos^2=1 by cos^2(theta).
  4. This gives tan^2(theta)+1=sec^2(theta).
  5. So sec^2(theta)=1+tan^2(theta).
  6. Check the result by substituting or estimating: the response should match 1 + tan^2(theta) and make sense in the original problem.

Answer: 1 + tan^2(theta)

4. If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).

Show solution
  1. Use sin^2+cos^2=1, so cos^2(theta)=1-9/25=16/25.
  2. cos(theta)=plus or minus 4/5.
  3. Quadrant I cosine is positive, so cos(theta)=4/5.

Answer: 4/5

5. If cos(theta)=8/17 and theta is in Quadrant IV, find sin(theta).

Show solution
  1. Use sin^2+cos^2=1, so sin^2(theta)=1-64/289=225/289.
  2. sin(theta)=plus or minus 15/17.
  3. Quadrant IV sine is negative, so sin(theta)=-15/17.

Answer: -15/17

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