Unit circle sine and cosine

• Interpret cosine and sine as the x and y coordinates of a point on the unit circle.
• Use a reference angle to determine coordinate magnitudes in any quadrant.
• Apply quadrant signs and the unit-circle identity to verify an exact coordinate.

Build the idea

Worked example

1. Step 1

   One hundred fifty degrees lies between ninety and one hundred eighty degrees, so its terminal point is in quadrant two with negative x and positive y.

   $$90^\circ<150^\circ<180^\circ$$

2. Step 2

   Measure the acute gap from the negative x-axis. Subtracting one hundred fifty from one hundred eighty gives a thirty-degree reference angle.

   $$180^\circ-150^\circ=30^\circ$$

3. Step 3

   A thirty-degree reference triangle has horizontal magnitude one half of the square root of three and vertical magnitude one half. Quadrant two makes only the horizontal value negative.

   $$(\cos150^\circ,\sin150^\circ)=\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$$

4. Step 4

   Square both coordinates and add. Their signs disappear after squaring, and the sum equals one, confirming that the point is exactly one unit from the origin.

   $$\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2=\frac{3}{4}+\frac{1}{4}=1$$

• Treating the reference angle itself as the original angle and assigning both trigonometric values positive signs.

  The reference angle supplies magnitudes only. Return to the original terminal side: in quadrant two the horizontal coordinate is negative and the vertical coordinate is positive.

• Swapping sine and cosine because the vertical coordinate is noticed first in the diagram.

  Keep the ordered-pair convention visible: x comes first and equals cosine, while y comes second and equals sine. Checking the point at zero degrees reinforces this order.

Find the exact cosine and sine of two hundred ten degrees using its reference angle and quadrant signs.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Attach coordinates to an angle

   Draw a circle of radius one centered at the origin. Rotate a radius from the positive x-axis through angle theta. Where the radius meets the circle, read the horizontal coordinate as cosine theta and the vertical coordinate as sine theta. This ordered pair definition works even when the angle is not acute.

   • The point P of theta has coordinates cosine theta, sine theta.
2. Part 2

   Recover familiar magnitudes

   Project the endpoint to the x-axis to form a right triangle. The radius is the hypotenuse and has length one. The acute reference angle connects the coordinate magnitudes to familiar triangle ratios. Direction remains separate: a projection left of the origin is negative, and a projection below the origin is negative.

   • Cosine squared theta plus sine squared theta equals one.
3. Part 3

   Use quadrant two

   One hundred fifty degrees ends in quadrant two. Its reference angle is thirty degrees because the terminal ray is thirty degrees short of one hundred eighty. The thirty-degree magnitudes are one half of the square root of three horizontally and one half vertically. Left makes cosine negative, while up keeps sine positive.

   • Ninety degrees is less than one hundred fifty degrees, which is less than one hundred eighty degrees.
   • One hundred eighty degrees minus one hundred fifty degrees equals thirty degrees.
   • The ordered pair cosine one hundred fifty degrees and sine one hundred fifty degrees equals the ordered pair negative one half of the square root of three, and one half.
4. Part 4

   Verify the endpoint

   Square the two coordinates and add them. Three fourths plus one fourth equals one, so the endpoint is one unit from the origin. This confirms the magnitudes, while the quadrant confirms the signs. Together they distinguish the correct point from three other points with the same reference triangle.

   • The quantity negative one half of the square root of three, squared, plus the quantity one half, squared, equals three fourths plus one fourth, which equals one.
   • Cosine squared theta plus sine squared theta equals one.

Scope and limitations

• This lesson focuses on degree measure and one special reference triangle; it does not develop radians, arbitrary-angle approximations, or inverse trigonometric functions.
• Static coordinate projections explain values but do not yet model sinusoidal graphs, angular velocity, or complex exponential interpretations.