Mathematics · 18 / 23
Unit circle and radians
High school · intermediate · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
On the unit circle, angle θ gives coordinates (cos θ, sin θ). One turn is 2π radians or 360°. Radians connect angle with unit-circle arc length. Sine and cosine have period 2π and quadrant-dependent signs. Pythagoras gives cos²θ+sin²θ=1.
On the unit circle, the point at angle θ has coordinates (cos θ, sin θ). The drawing shows π/4; projections give the coordinates.
- 01Radians
- 02Circle point
- 03Cosine x / sine y
- 04Period and sign
WORKED EXAMPLE
From principle to reasoning
π/6 is 30°, with cosine √3/2 and sine 1/2. At π the point is (−1,0). Studio inputs use radians.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Find sin(π/2) and cos(π/2).
One hint
Locate the top of the unit circle.
Reveal and understand the solution
Coordinates (0,1) give cosine 0 and sine 1.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Convert 150° to radians.
A hint to get started
180° equals π radians.
Step-by-step solution
- Multiply 150 by π/180.
- Simplify 150/180 to 5/6.
- 150° = 5π/6 radians.
Which step did you miss? Retry without help, then explain why the method works.
Find cos(π/3), sin(π/3) and check cos² + sin².
A hint to get started
The point lies in the first quadrant.
Step-by-step solution
- cos(π/3) = 1/2 and sin(π/3) = √3/2.
- Their squares are 1/4 and 3/4.
- Their sum is 1, as for every point on the unit circle.
Which step did you miss? Retry without help, then explain why the method works.
Solve sin(x) = 1/2 in [0,2π).
A hint to get started
A horizontal line intersects the circle twice.
Step-by-step solution
- The reference angle is π/6.
- The other angle is π − π/6 = 5π/6.
- The two solutions are π/6 and 5π/6.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Do not mix degrees and radians. cos²θ means (cos θ)².
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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