Trigonometry · Unit Circle & Waves
Link a rotating unit-circle radius to exact sine, cosine, quadrants, phase and frequency of their periodic graphs.
NCERT alignment: Class 11; chapter Trigonometric Functions. Exam practice: JEE Main, JEE Advanced.
Learning objectives
Students should use the live simulator to test a quantitative prediction, read measurements and explain the limits of the model. Do not rely on the animation alone: calculate the expected result before adjusting controls.
Theory, equations and interpretations
The unit-circle definition
On a unit circle, the horizontal projection is cos α and the vertical projection is sin α. α = nθ+φ is the effective angle.
Important relation: (cos α, sin α); cos²α + sin²α = 1
Radians and degrees
One full revolution is 360°=2π radians. A sine or cosine curve has period 360°/n with integer positive n.
Important relation: α(rad) = α(°) × π/180
Amplitude and phase
A sets graph height; φ shifts the graph horizontally by −φ/n degrees. These graph transforms do not change the unit-circle radius.
Important relation: y=A sin(nθ+φ)
Signs by quadrant
In quadrant II, sine is positive while cosine is negative. Compare signs by dragging the angle around the circle.
Important relation: sin(π−α)=sin α; cos(π−α)=−cos α
Practical investigations
- Exact special angles: Set n=1,φ=0,A=1; check θ=0°,30°,45°,60°,90° against known values.
- Investigate signs: Move θ into quadrants II, III and IV and record the signs of sine and cosine.
- Frequency and period: Change n from 1 to 3. How many complete oscillations fit in 360°?
- Phase equivalence: Show graph sin(θ+90°) matches cos θ at n=1,A=1.
Worked exam-style questions and answers
Q1. sin 30° is
Options: A. 1; B. 1/2; C. √3/2; D. 0
Answer: B. In the unit circle sin 30°=1/2.
Q2. cos 120° is
Options: A. 1/2; B. −1/2; C. −√3/2; D. 1
Answer: B. cos(180°−60°)=−cos 60°=−1/2.
Q3. Period of sin(3θ), θ in degrees, is
Options: A. 360°; B. 180°; C. 120°; D. 90°
Answer: C. Period=360°/3=120°.
Q4. At θ=270°, sin θ equals
Options: A. −1; B. 0; C. 1; D. 1/2
Answer: A. The unit-circle point at 270° is (0,−1).
Q5. sin²θ+cos²θ is
Options: A. −1; B. 0; C. 1; D. 2
Answer: C. Pythagorean identity from x²+y²=1.
Q6. sin(θ+90°) equals
Options: A. −cos θ; B. sin θ; C. cos θ; D. −sin θ
Answer: C. Angle addition: sin(θ+90°)=cos θ.
Common misconceptions and limitations
Angles are entered in degrees and converted to radians internally. The unit circle is always radius 1; the separate amplitude parameter scales only the waveform, not the geometric unit circle.
All learning content is available free of charge. The simulation is a simplified educational model. Check assumptions and units before applying a result. NANDAX SimStudio is not affiliated with NCERT or examination boards.
Teacher classroom use
Ask students to form a hypothesis, vary one parameter at a time, keep a record of graph readings, explain any change and compare results with the analytic relation. The on-screen worksheet offers questions with an answer key.