Pythagorean Identity Quiz: Pythagorean Identity

  • 10th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) Which identity holds for every real angle θ?

Explanation

On the unit circle a point at angle θ is (cosθ, sinθ). The radius is 1, so by the distance formula x^2 + y^2 = 1. Substituting x = cosθ and y = sinθ gives cos^2θ + sin^2θ = 1.

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About This Quiz
Pythagorean Identity Quiz: Pythagorean Identity - Quiz

How well do you know the most famous identity in trigonometry? This quiz takes you through the Pythagorean Identity and shows how it links sine and cosine. You’ll test simple expressions, verify relationships, and see how the identity appears in many trig problems. Jump in and see how easily it... see moreall comes together.
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2) On the unit circle, the coordinates are (cosθ, sinθ). What is cos^2θ + sin^2θ?

Explanation

The unit circle has radius 1, so the distance from the origin to (cosθ, sinθ) is √(cos^2θ + sin^2θ) = 1. Squaring both sides gives cos^2θ + sin^2θ = 1.

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3) For any real θ, sin^2θ + cos^2θ = 1 is always true.

Explanation

This is an identity derived from x^2 + y^2 = 1 on the unit circle with x = cosθ and y = sinθ; it holds for all real θ.

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4) If sinθ = 3/5 and 0 ≤ θ ≤ π/2, then cosθ = ________.

Explanation

Use sin^2θ + cos^2θ = 1. Compute cos^2θ = 1 − sin^2θ = 1 − (3/5)^2 = 1 − 9/25 = 16/25, so cosθ = √(16/25) = 4/5 (positive in Quadrant I).

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5) If cosθ = −12/13 and θ is in Quadrant II, what is sinθ?

Explanation

Compute sin^2θ = 1 − cos^2θ = 1 − (144/169) = 25/169 ⇒ |sinθ| = 5/13. In Quadrant II, sinθ > 0 and cosθ

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6) Using the identity, express cos^2θ in terms of sinθ.

Explanation

Rearrange sin^2θ + cos^2θ = 1 to get cos^2θ = 1 − sin^2θ.

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7) Select all identities that follow from sin^2θ + cos^2θ = 1.

Explanation

From sin^2θ + cos^2θ = 1 we get 1 − cos^2θ = sin^2θ and 1 − sin^2θ = cos^2θ. Dividing by cos^2θ (cosθ ≠ 0) gives (sin^2θ/cos^2θ) + 1 = 1/cos^2θ ⇒ tan^2θ + 1 = sec^2θ. The equation sinθ + cosθ = 1 is not an identity.

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8) For any real θ, sin^2θ = 1 − cos^2θ.

Explanation

Rearranging sin^2θ + cos^2θ = 1 gives sin^2θ = 1 − cos^2θ, valid for all real θ.

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9) If a point on the unit circle is (−√3/2, 1/2), what is sin^2θ + cos^2θ?

Explanation

Compute: cos^2θ + sin^2θ = ( (−√3/2)^2 ) + ( (1/2)^2 ) = (3/4) + (1/4) = 1.

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10) In a right triangle with hypotenuse 10, opposite 6, and adjacent 8, sin^2θ + cos^2θ = ________.

Explanation

sinθ = opp/hyp = 6/10 = 3/5, cosθ = adj/hyp = 8/10 = 4/5. Then sin^2θ + cos^2θ = (3/5)^2 + (4/5)^2 = 9/25 + 16/25 = 25/25 = 1.

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11) If sinθ = −0.8 and θ is in Quadrant III, what is cosθ?

Explanation

Compute cos^2θ = 1 − sin^2θ = 1 − 0.64 = 0.36 ⇒ |cosθ| = 0.6. In Quadrant III, both sine and cosine are negative, so cosθ = −0.6.

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12) Which expression is true for any real θ?

Explanation

From cos^2θ = 1 − sin^2θ, we take square roots to get cosθ = ±√(1 − sin^2θ). The sign depends on θ’s quadrant.

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13) Choose all angles for which sin^2θ + cos^2θ = 1 holds.

Explanation

sin^2θ + cos^2θ = 1 is an identity for all real θ. It is not restricted to any quadrant.

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14) On the unit circle, the distance from (0,0) to (cosθ, sinθ) is 1 for every θ.

Explanation

By definition of the unit circle, every point (cosθ, sinθ) lies at radius 1 from the origin, so √(cos^2θ + sin^2θ) = 1.

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15) If cosθ = 0, what is sin^2θ?

Explanation

Use sin^2θ + cos^2θ = 1. If cosθ = 0, then sin^2θ = 1 − 0 = 1, meaning sinθ = ±1 at θ = π/2 + kπ.

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16) Complete the unit circle equation: For (x, y) = (cosθ, sinθ), x^2 + y^2 = ________.

Explanation

The unit circle has radius 1, so x^2 + y^2 = 1. Substituting x = cosθ, y = sinθ yields cos^2θ + sin^2θ = 1.

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17) Evaluate sin^2(π/6) + cos^2(π/6).

Explanation

sin(π/6) = 1/2 ⇒ sin^2 = 1/4; cos(π/6) = √3/2 ⇒ cos^2 = 3/4; sum = 1/4 + 3/4 = 1.

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18) If sinθ = 7/25 and θ is acute, what is cosθ?

Explanation

cos^2θ = 1 − sin^2θ = 1 − 49/625 = 576/625 ⇒ cosθ = √(576/625) = 24/25 (positive for acute angles).

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19) Select all pairs (sinθ, cosθ) that are possible for some real θ.

Explanation

Check sin^2 + cos^2: A) 9/25 + 16/25 = 1 ✓; B) 144/169 + 25/169 = 1 ✓; C) 1/2 + 1/2 = 1 ✓; D) 1.44 + 0 > 1 and |sin| ≤ 1 is violated ✗; E) 16/25 + 9/25 = 1 ✓.

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20) There exists a real θ such that sin^2θ + cos^2θ = 0.

Explanation

Since sin^2θ ≥ 0 and cos^2θ ≥ 0 for all θ and their sum is identically 1, it can never be 0. Therefore no real θ satisfies the equation.

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Which identity holds for every real angle θ?
On the unit circle, the coordinates are (cosθ, sinθ). What is...
For any real θ, sin^2θ + cos^2θ = 1 is always true.
If sinθ = 3/5 and 0 ≤ θ ≤ π/2, then cosθ = ________.
If cosθ = −12/13 and θ is in Quadrant II, what is sinθ?
Using the identity, express cos^2θ in terms of sinθ.
Select all identities that follow from sin^2θ + cos^2θ = 1.
For any real θ, sin^2θ = 1 − cos^2θ.
If a point on the unit circle is (−√3/2, 1/2), what is sin^2θ +...
In a right triangle with hypotenuse 10, opposite 6, and adjacent 8,...
If sinθ = −0.8 and θ is in Quadrant III, what is cosθ?
Which expression is true for any real θ?
Choose all angles for which sin^2θ + cos^2θ = 1 holds.
On the unit circle, the distance from (0,0) to (cosθ, sinθ) is 1 for...
If cosθ = 0, what is sin^2θ?
Complete the unit circle equation: For (x, y) = (cosθ, sinθ), x^2 +...
Evaluate sin^2(π/6) + cos^2(π/6).
If sinθ = 7/25 and θ is acute, what is cosθ?
Select all pairs (sinθ, cosθ) that are possible for some real θ.
There exists a real θ such that sin^2θ + cos^2θ = 0.
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