How can I use the Pythagorean identity to find a missing sine or cosine value?
Quick answer
Start with sin²θ + cos²θ = 1. If sine is known, calculate cos θ = ±√(1 − sin²θ); if cosine is known, calculate sin θ = ±√(1 − cos²θ). The square root gives a magnitude, so use the angle’s quadrant to choose the sign: sine matches the unit-circle y-coordinate and cosine the x-coordinate. Finally, substitute both values into the identity and check that their squares add to 1. [S1]
The question, the details, the next step.
I know one value, but I’m not sure how to rearrange the identity. What should I do first?
Write the identity as sin²θ + cos²θ = 1, then isolate the unknown squared value. For a known sine, subtract its square from 1: cos²θ = 1 − sin²θ. Take a square root to get cos θ = ±√(1 − sin²θ). For a known cosine, the matching rearrangement is sin θ = ±√(1 − cos²θ). [S1]
The ± matters: taking a square root finds the size of the missing value but leaves two possible signs. Don’t decide the sign from the square-root calculation alone. Keep the known value and the missing value attached to the same angle θ throughout the calculation.
How do I know whether to use the positive or negative result?
Use the quadrant containing θ. On the unit circle, cosine is the x-coordinate and sine is the y-coordinate, so the signs come from whether that coordinate is positive or negative. [S1]
For a quick sign guide:
- Quadrant I: sine positive, cosine positive.
- Quadrant II: sine positive, cosine negative.
- Quadrant III: sine negative, cosine negative.
- Quadrant IV: sine negative, cosine positive.
If you’re only told one ratio and not given the angle’s quadrant or equivalent sign information, there may be two possible answers. Keep both possibilities until you have enough information to choose. The quadrant identifies the sign of each ratio; it does not change the square-root calculation.
Could you show a complete example where the answer is negative?
Suppose sin θ = 3/5 and θ is in Quadrant II. First find the cosine magnitude:
cos θ = ±√(1 − (3/5)²) = ±√(1 − 9/25) = ±√(16/25) = ±4/5.
In Quadrant II, cosine is negative, so cos θ = −4/5. Now check the original identity:
(3/5)² + (−4/5)² = 9/25 + 16/25 = 1.
The check confirms the pair fits the identity; the quadrant information is what selected the negative sign. Notice that squaring −4/5 in the check gives a positive 16/25, so the check alone would not distinguish +4/5 from −4/5.
What if cosine is the value I know, and how can I avoid sign mistakes?
Suppose cos θ = −5/13 and θ is in Quadrant III. Rearrange for sine and calculate:
sin θ = ±√(1 − (−5/13)²) = ±√(1 − 25/169) = ±√(144/169) = ±12/13.
Quadrant III has negative sine, so sin θ = −12/13. Check: (−12/13)² + (−5/13)² = 144/169 + 25/169 = 1.
A reusable routine is: square the known ratio; subtract from 1; take the square root; choose the sign from the quadrant; substitute both values back into the identity. Keep parentheses around a negative known value when squaring it. In particular, squaring −5/13 makes the squared term positive, not negative.
A practical checklist
Write sin²θ + cos²θ = 1 and identify which ratio is missing.
Rearrange for the missing squared ratio, then calculate the square root.
Use the quadrant to choose the sine or cosine sign.
Substitute both values into the original identity and verify the squares sum to 1.
A few more useful answers
Why does the square root give two possible signs?
A positive and negative number have the same square, so the identity determines the missing value’s magnitude but not its sign. Use the quadrant or other angle information to choose.
Can I find the missing sign if I’m given only the known ratio?
Not always. The same sine or cosine value can occur at angles in different quadrants, so you may need the quadrant or another condition to decide between signs.
What does the identity check confirm?
It confirms that the two proposed values have squares that add to 1. By itself, that check does not determine the angle’s quadrant; choose signs using the angle information first.
Sources
References used for this guide.
- [S1] OpenStax Precalculus 2e: Chapter 5 Key Concepts openstax.org · Checked