Double-Angle Formulas: Simplify & Evaluate Quiz

  • 11th Grade
Reviewed by Cierra Henderson
Cierra Henderson, MBA |
K-12 Expert
Review Board Member
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.
 , MBA
By Thames
T
Thames
Community Contributor
Quizzes Created: 8156 | Total Attempts: 9,586,195
| Attempts: 19 | Questions: 20 | Updated: Jan 22, 2026
Please wait...
Question 1 / 21
🏆 Rank #-- ▾
Score 0/100
1) Simplify sin(2θ) + cos(2θ) when θ = 0.

Explanation

Given: θ = 0. Goal: compute.

Step 1: sin(0) = 0 ⇒ sin(2·0) = 0.

Step 2: cos(0) = 1 ⇒ cos(2·0) = 1.

Step 3: Sum = 1.

So, the final answer is 1.

Submit
Please wait...
About This Quiz
Double-angle Formulas: Simplify & Evaluate Quiz - Quiz

Now that you know the double-angle formulas, it’s time to apply them! This quiz focuses on simplifying and evaluating expressions using double-angle identities. You’ll use relationships like cos(2θ) = 1 − 2sin²θ and tan(2θ) = (2tanθ)/(1 − tan²θ) to find exact values of trig functions. Each question helps you see... see morehow double-angle identities make trigonometric evaluation quicker and more efficient — even without a calculator.
see less

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) Which identity is true for all θ?

Explanation

Given: identities. Goal: pick true one.

Step 1: Standard: sin(2θ) = 2 sinθ cosθ.

So, the final answer is sin(2θ) = 2 sinθ cosθ.

Submit
3) If cos(2θ) = −1, find θ (radians).

Explanation

Given: cos(2θ) = −1. Goal: solve for θ.

Step 1: 2θ = π + 2πn ⇒ θ = π/2 + πn.

So, the final answer is π/2 + πn.

Submit
4) Simplify sin(2θ) + cos(2θ) when θ = 30°.

Explanation

Given: θ = 30°. Goal: compute.

Step 1: sin(2θ) = sin60° = √3/2; cos(2θ) = cos60° = 1/2.

Step 2: Sum = (√3 + 1)/2.

So, the final answer is (√3 + 1)/2.

Submit
5) Simplify sin(2θ) when sinθ = 4/5 and θ is in Quadrant II.

Explanation

Given: sinθ = 4/5 (QII ⇒ cosθ < 0). Goal: sin(2θ).

Step 1: cosθ = −3/5.

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

So, the final answer is −24/25.

Submit
6) Simplify cos(2θ) in terms of sinθ.

Explanation

Given: express cos(2θ) via sinθ. Goal: pick a correct form.

Step 1: cos(2θ) = 1 − 2 sin²θ.

So, the final answer is 1 − 2 sin²θ.

Submit
7) If tanθ = 1/3, find tan(2θ).

Explanation

Given: tanθ = 1/3. Goal: tan(2θ).

Step 1: tan(2θ) = (2t)/(1 − t²) = (2/3)/(1 − 1/9) = (2/3)/(8/9) = 3/4.

So, the final answer is 3/4.

Submit
8) Which expression equals sin(2θ)/cos(2θ)?

Explanation

Given: quotient of double-angles. Goal: simplify.

Step 1: sin(2θ)/cos(2θ) = tan(2θ).

So, the final answer is tan(2θ).

Submit
9) Given sinθ = 12/13, compute cos(2θ).

Explanation

Given: sinθ = 12/13. Goal: cos(2θ).

Step 1: cos(2θ) = 1 − 2 sin²θ = 1 − 2·(144/169) = (169 − 288)/169 = −119/169.

So, the final answer is −119/169.

Submit
10) Simplify 1 − 2 cos²θ.

Explanation

Given: 1 − 2 cos²θ. Goal: rewrite.

Step 1: cos(2θ) = 2 cos²θ − 1 ⇒ 1 − 2 cos²θ = −(2 cos²θ − 1) = −cos(2θ).

So, the final answer is −cos(2θ).

Submit
11) If tanθ = 2 and θ is acute, find cos(2θ).

Explanation

Given: tanθ = 2. Goal: cos(2θ).

Step 1: cos(2θ) = (1 − tan²θ)/(1 + tan²θ) = (1 − 4)/(1 + 4) = −3/5.

So, the final answer is −3/5.

Submit
12) Express sin²θ in terms of cos(2θ).

Explanation

Given: power-reduction. Goal: sin²θ.

Step 1: sin²θ = (1 − cos(2θ))/2.

So, the final answer is (1 − cos(2θ))/2.

Submit
13) Simplify tan(2θ) if tanθ = 3/4.

Explanation

Given: t = 3/4. Goal: tan(2θ).

Step 1: tan(2θ) = (2t)/(1 − t²) = (2·3/4)/(1 − 9/16) = (3/2)/(7/16) = 24/7.

So, the final answer is 24/7.

Submit
14) If sinθ = 8/17, find sin(2θ).

Explanation

Given: sinθ = 8/17. Goal: sin(2θ).

Step 1: cosθ = √(1 − 64/289) = 15/17 (taking the principal value).

Step 2: sin(2θ) = 2 sinθ cosθ = 2·(8/17)·(15/17) = 240/289.

So, the final answer is 240/289.

Submit
15) Simplify cos(2θ) + sin(2θ) when sinθ = 3/5, cosθ = 4/5.

Explanation

Given: sinθ = 3/5, cosθ = 4/5. Goal: sum of double-angles.

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

Step 2: cos(2θ) = (4/5)² − (3/5)² = 16/25 − 9/25 = 7/25.

Step 3: Sum = 31/25.

So, the final answer is 31/25.

Submit
16) If cos(2θ) = 0, what is θ (degrees)?

Explanation

Given: cos(2θ) = 0. Goal: solutions for θ.

Step 1: 2θ = 90° + 180°n ⇒ θ = 45° + 90°n.

So, the final answer is 45° + 90°n.

Submit
17) Simplify sin(2θ) + cos(2θ) when θ = 45°.

Explanation

Given: θ = 45°. Goal: evaluate.

Step 1: sin(2θ) = sin90° = 1; cos(2θ) = cos90° = 0.

Step 2: Sum = 1.

So, the final answer is 1.

Submit
18) Express tan(2θ) in terms of tanθ.

Explanation

Given: double-angle for tangent. Goal: formula.

Step 1: tan(2θ) = (2 tanθ)/(1 − tan²θ).

So, the final answer is (2 tanθ)/(1 − tan²θ).

Submit
19) Simplify 1 − cos(2θ).

Explanation

Given: 1 − cos(2θ). Goal: reduce.

Step 1: cos(2θ) = 1 − 2 sin²θ ⇒ 1 − cos(2θ) = 2 sin²θ.

So, the final answer is 2 sin²θ.

Submit
20) If sinθ = 5/13 and θ in Quadrant I, find tan(2θ).

Explanation

Given: sinθ = 5/13, QI. Goal: tan(2θ).

Step 1: cosθ = 12/13 ⇒ tanθ = 5/12.

Step 2: tan(2θ) = (2·5/12)/(1 − 25/144) = (5/6)/(119/144) = 120/119.

So, the final answer is 120/119.

Submit
×
Saved
Thank you for your feedback!
View My Results
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.
Cancel
  • All
    All (20)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Simplify sin(2θ) + cos(2θ) when θ = 0.
Which identity is true for all θ?
If cos(2θ) = −1, find θ (radians).
Simplify sin(2θ) + cos(2θ) when θ = 30°.
Simplify sin(2θ) when sinθ = 4/5 and θ is in...
Simplify cos(2θ) in terms of sinθ.
If tanθ = 1/3, find tan(2θ).
Which expression equals sin(2θ)/cos(2θ)?
Given sinθ = 12/13, compute cos(2θ).
Simplify 1 − 2 cos²θ.
If tanθ = 2 and θ is acute, find cos(2θ).
Express sin²θ in terms of cos(2θ).
Simplify tan(2θ) if tanθ = 3/4.
If sinθ = 8/17, find sin(2θ).
Simplify cos(2θ) + sin(2θ) when sinθ = 3/5,...
If cos(2θ) = 0, what is θ (degrees)?
Simplify sin(2θ) + cos(2θ) when θ = 45°.
Express tan(2θ) in terms of tanθ.
Simplify 1 − cos(2θ).
If sinθ = 5/13 and θ in Quadrant I, find tan(2θ).
play-Mute sad happy unanswered_answer up-hover down-hover success oval cancel Check box square blue
Alert!