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Unit Circle (Full Visual Map)
All Angles, Exact Values, and Quadrant Signs
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The unit circle is a circle of radius 1 centered at the origin, and it is one of the most important visual tools in trigonometry. It connects angles, coordinates, and trig functions in a single diagram. By learning the unit circle, students can quickly find exact values of sine, cosine, and tangent for many common angles. It also helps build a foundation for graphing periodic functions and solving equations.
Each point on the unit circle corresponds to an angle measured from the positive x-axis, and that point has coordinates (cos theta, sin theta). This means cosine gives the horizontal position and sine gives the vertical position. Tangent can then be found from tan theta = sin theta / cos theta when cos theta is not zero. The signs of these values change by quadrant, so the full circle acts like a map of trig behavior.
Key Facts
- Unit circle equation: x^2 + y^2 = 1
- For an angle theta, the point on the circle is (cos theta, sin theta)
- tan theta = sin theta / cos theta
- 180 degrees = pi radians
- Reference angle helps find exact trig values using related acute angles
- Quadrant signs: I (+,+), II (-,+), III (-,-), IV (+,-) for (cos theta, sin theta)
Vocabulary
- Unit circle
- A circle with radius 1 centered at the origin of the coordinate plane.
- Radians
- A way to measure angles based on arc length, where a full circle is 2pi radians.
- Reference angle
- The acute angle formed between the terminal side of an angle and the x-axis.
- Terminal side
- The final position of a ray after it rotates from its starting side to form an angle.
- Quadrant
- One of the four regions of the coordinate plane created by the x-axis and y-axis.
Common Mistakes to Avoid
- Mixing up sine and cosine, which is wrong because on the unit circle cosine is the x-coordinate and sine is the y-coordinate. Always read points as (cos theta, sin theta).
- Using degree values in place of radian values without converting, which is wrong because many unit circle formulas and labels are written in radians. Use 180 degrees = pi radians to convert correctly.
- Forgetting signs in different quadrants, which is wrong because the same reference angle has different coordinate signs depending on location. Check whether x and y should be positive or negative before writing trig values.
- Computing tangent when cosine equals zero, which is wrong because tan theta = sin theta / cos theta and division by zero is undefined. At angles like pi/2 and 3pi/2, tangent does not exist.
Practice Questions
- 1 Find the coordinates on the unit circle for theta = 5pi/6, and then give sin theta, cos theta, and tan theta.
- 2 Convert 225 degrees to radians, then state the unit circle coordinates for that angle.
- 3 Angles 30 degrees and 150 degrees have the same reference angle. Explain why their sine values are equal but their cosine values have opposite signs.