Double-Angle Applications with Pythagorean Identity Quiz

  • 10th Grade
Reviewed by Cierra Henderson
Cierra Henderson, MBA |
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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Quizzes Created: 8156 | Total Attempts: 9,586,195
| Attempts: 12 | Questions: 20 | Updated: Jan 22, 2026
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1) If sinθ = 3/5 and θ is in Quadrant II, find cosθ.

Explanation

Given: sinθ = 3/5 (QII). Goal: cosθ.

Step 1: cosθ = −√(1 − (3/5)²) = −√(16/25) = −4/5.

So, the final answer is −4/5.

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About This Quiz
Double-angle Applications With Pythagorean Identity Quiz - Quiz

In this quiz, you’ll connect double-angle formulas to geometry, triangles, and the unit circle. You’ll apply sin(2θ) and cos(2θ) to find missing trig values, model angles on the coordinate plane, and explore how these identities describe real-world motion and rotation. By combining your knowledge of Pythagorean and double-angle relationships, you’ll... see morestrengthen your understanding of how trigonometric formulas work together in problem solving.
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2) If cosθ = −7/25 and θ is in Quadrant III, find sinθ.

Explanation

Given: cosθ = −7/25 (QIII). Goal: sinθ.

Step 1: sinθ = −√(1 − 49/625) = −√(576/625) = −24/25.

So, the final answer is −24/25.

Submit
3) Given sinθ = −12/13 and θ is in Quadrant IV, compute cosθ.

Explanation

Given: sinθ negative, QIV ⇒ cosθ > 0. Goal: cosθ.

Step 1: cosθ = √(1 − 144/169) = 5/13.

So, the final answer is 5/13.

Submit
4) If cosθ = √3/2 and θ is in Quadrant I, find sinθ.

Explanation

Given: cosθ = √3/2 (QI). Goal: sinθ.

Step 1: sinθ = √(1 − 3/4) = 1/2.

So, the final answer is 1/2.

Submit
5) If sinθ = 5/13, find cosθ assuming θ is in Quadrant I.

Explanation

Given: sinθ = 5/13 (QI). Goal: cosθ.

Step 1: cosθ = √(1 − 25/169) = 12/13.

So, the final answer is 12/13.

Submit
6) A point P on the unit circle has y-coordinate −0.6. Find the possible x-coordinate(s).

Explanation

Given: y = sinθ = −0.6. Goal: x = cosθ.

Step 1: x = ±√(1 − 0.36) = ±√0.64 = ±0.8.

So, the final answer is ±0.8.

Submit
7) If sinθ = 0.8 and θ is in Quadrant II, compute cosθ.

Explanation

Given: sinθ = 0.8 (QII ⇒ cosθ < 0). Goal: cosθ.

Step 1: cosθ = −√(1 − 0.64) = −0.6.

So, the final answer is −0.6.

Submit
8) For cosθ = −√5/3 with θ in Quadrant II, determine sinθ.

Explanation

Given: cosθ negative, QII ⇒ sinθ > 0. Goal: sinθ.

Step 1: sinθ = √(1 − 5/9) = √(4/9) = 2/3.

So, the final answer is 2/3.

Submit
9) If x = −9/10 and θ is in Quadrant II, find y.

Explanation

Given: x = cosθ = −9/10 (QII ⇒ y > 0). Goal: y.

Step 1: y = √(1 − 81/100) = √(19/100) = √19/10.

So, the final answer is √19/10.

Submit
10) If cosθ = 4/5 and sinθ > 0, find sin(2θ).

Explanation

Given: cosθ = 4/5, sinθ > 0 ⇒ sinθ = 3/5. Goal: sin(2θ).

Step 1: sin(2θ) = 2·(3/5)·(4/5) = 24/25.

So, the final answer is 24/25.

Submit
11) Given sinθ = −3/5, determine cos(2θ).

Explanation

Given: sinθ = −3/5. Goal: cos(2θ).

Step 1: cos(2θ) = 1 − 2·(9/25) = 1 − 18/25 = 7/25.

So, the final answer is 7/25.

Submit
12) If cosθ = −5/13 and θ is in Quadrant II, evaluate sin(2θ).

Explanation

Given: cosθ = −5/13 (QII ⇒ sinθ > 0). Goal: sin(2θ).

Step 1: sinθ = 12/13.

Step 2: sin(2θ) = 2·(12/13)·(−5/13) = −120/169.

So, the final answer is −120/169.

Submit
13) Right triangle with hypotenuse 13 and adjacent side to θ is 5. Find sinθ, then cos(2θ).

Explanation

Given: adjacent = 5, hypotenuse = 13. Goal: sinθ, cos(2θ).

Step 1: opposite = √(13² − 5²) = 12 ⇒ sinθ = 12/13.

Step 2: cos(2θ) = 1 − 2 sin²θ = 1 − 2·(144/169) = −119/169.

So, the final answer is 12/13 and −119/169.

Submit
14) On the unit circle, if sinθ = t (t ∈ [−1, 1]), which expression for cosθ is always valid?

Explanation

Given: sin²θ + cos²θ = 1. Goal: cosθ in terms of t.

Step 1: cosθ = ±√(1 − t²), sign by quadrant.

So, the final answer is ±√(1 − t²).

Submit
15) If sinθ = −√7/4 and θ is in Quadrant IV, find cosθ.

Explanation

Given: sinθ negative, QIV ⇒ cosθ > 0. Goal: cosθ.

Step 1: cosθ = √(1 − 7/16) = √(9/16) = 3/4.

So, the final answer is 3/4.

Submit
16) If cosθ = 1/3 and θ is in Quadrant I, find sinθ and tanθ.

Explanation

Given: cosθ = 1/3 (QI). Goal: sinθ, tanθ.

Step 1: sinθ = √(1 − 1/9) = √(8/9) = 2√2/3.

Step 2: tanθ = sinθ/cosθ = (2√2/3)/(1/3) = 2√2.

So, the final answer is sinθ = 2√2/3, tanθ = 2√2.

Submit
17) A unit direction vector has x-component 2/√5 (QI). Find sinθ.

Explanation

Given: cosθ = 2/√5, QI. Goal: sinθ.

Step 1: sinθ = √(1 − 4/5) = √(1/5) = 1/√5.

So, the final answer is 1/√5.

Submit
18) If sinθ = −4/5 and θ is in Quadrant III, find cosθ and then sin(2θ).

Explanation

Step 1: cosθ = −3/5.

Step 2: sin(2θ) = 2·(−4/5)·(−3/5) = 24/25.

So, the final answer is cosθ = −3/5, sin(2θ) = 24/25.

Submit
19) A unit-circle point has (x, y) with y = 5/13 and x < 0. Compute cos(2θ).

Explanation

Given: y = sinθ = 5/13, x < 0 ⇒ cosθ = −12/13. Goal: cos(2θ).

Step 1: cos(2θ) = cos²θ − sin²θ = (144 − 25)/169 = 119/169.

So, the final answer is 119/169.

Submit
20) If cosθ = −√2/2 and θ is in Quadrant II, find sinθ and cos(2θ).

Explanation

Given: cosθ = −√2/2, QII ⇒ sinθ > 0. Goal: sinθ, cos(2θ).

Step 1: sinθ = √2/2.

Step 2: cos(2θ) = cos²θ − sin²θ = 1/2 − 1/2 = 0.

So, the final answer is sinθ = √2/2, cos(2θ) = 0.

Submit
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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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If sinθ = 3/5 and θ is in Quadrant II, find cosθ.
If cosθ = −7/25 and θ is in Quadrant III, find...
Given sinθ = −12/13 and θ is in Quadrant IV, compute...
If cosθ = √3/2 and θ is in Quadrant I, find...
If sinθ = 5/13, find cosθ assuming θ is in Quadrant...
A point P on the unit circle has y-coordinate −0.6. Find the...
If sinθ = 0.8 and θ is in Quadrant II, compute cosθ.
For cosθ = −√5/3 with θ in Quadrant II,...
If x = −9/10 and θ is in Quadrant II, find y.
If cosθ = 4/5 and sinθ > 0, find sin(2θ).
Given sinθ = −3/5, determine cos(2θ).
If cosθ = −5/13 and θ is in Quadrant II, evaluate...
Right triangle with hypotenuse 13 and adjacent side to θ is 5....
On the unit circle, if sinθ = t (t ∈ [−1, 1]), which...
If sinθ = −√7/4 and θ is in Quadrant IV, find...
If cosθ = 1/3 and θ is in Quadrant I, find sinθ and...
A unit direction vector has x-component 2/√5 (QI). Find...
If sinθ = −4/5 and θ is in Quadrant III, find...
A unit-circle point has (x, y) with y = 5/13 and x < 0. Compute...
If cosθ = −√2/2 and θ is in Quadrant II, find...
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