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: 9,588,805
| 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)

What first name or nickname would you like us to use?

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!