Tags: unit circle, sine, cosine, tangent, standard position, terminal ray, coordinates on a circle, radius, radian measure, reference angle, AP Precalculus Unit 3
Sine, cosine, and tangent connect angles to coordinates on a circle. For any angle θ in standard position and a circle of radius r, the point where the terminal ray meets the circle is (r cos θ, r sin θ). When the circle is not a unit circle, you divide coordinates by r to get the trig ratios. The line y = 2x gives you a direction, and you use the Pythagorean theorem to extract the ratios.
Standard position
An angle whose vertex is at the origin, initial side along the positive x-axis, and terminal side (ray) rotated counterclockwise for positive angles, clockwise for negative.
Terminal ray
The ray that defines the "end" of the angle. The point where it intersects a circle centred at the origin gives you the coordinates used for trig ratios.
Unit circle
A circle centred at the origin with radius 1. On the unit circle, a point at angle θ has coordinates (cos θ, sin θ) directly.
Reference angle
The acute angle between the terminal ray and the x-axis. Used to find trig values in any quadrant by applying the correct sign.
Sine (sin θ)
For a point (x, y) on a circle of radius r: sin θ = y / r. On the unit circle, sin θ = y.
Cosine (cos θ)
For a point (x, y) on a circle of radius r: cos θ = x / r. On the unit circle, cos θ = x.
Tangent (tan θ)
tan θ = y / x = sin θ / cos θ. Undefined when x = 0 (i.e. at θ = π/2 and θ = 3π/2).
Given a point (x, y) on a circle centred at the origin:
Calculate the radius: r = √(x² + y²)
sin θ = y / r
cos θ = x / r
tan θ = y / x
Example: the terminal ray passes through (−4.5, 1.25). r = √(4.5² + 1.25²) = √(20.25 + 1.5625) = √21.8125 ≈ 4.669. sin θ = 1.25 / 4.669, cos θ = −4.5 / 4.669, tan θ = 1.25 / (−4.5) = −5/18.
The coordinates of the point P where the terminal ray at angle θ intersects a circle of radius r are:
P = (r cos θ, r sin θ)
Example: θ = 4π/3, r = 6. The reference angle is 4π/3 − π = π/3. In the third quadrant, both x and y are negative. cos(π/3) = 1/2, sin(π/3) = √3/2. So P = (6 · (−1/2), 6 · (−√3/2)) = (−3, −3√3).
If you know the angle θ and the coordinates of point P on the circle, you can find r directly:
r = √(x² + y²)
Example: θ = π/3, P = (√3/2, 3/2). r = √((√3/2)² + (3/2)²) = √(3/4 + 9/4) = √(12/4) = √3.
You can verify: r cos(π/3) = √3 · (1/2) = √3/2 ✓ and r sin(π/3) = √3 · (√3/2) = 3/2 ✓.
When the terminal ray coincides with y = 2x in the first quadrant:
Pick any convenient point on the line, say (1, 2).
r = √(1² + 2²) = √5.
sin θ = 2/√5 = 2√5/5
cos θ = 1/√5 = √5/5
tan θ = 2/1 = 2
The quadrant tells you the signs. In the first quadrant, all values are positive.
For standard angles on a circle of radius r:
At π/3: the point is (r/2, r√3/2)
At 5π/3: the point is (r/2, −r√3/2)
The segment between two such points (say C and D on the same circle) can be found using the distance formula.
Example: circle of radius 3, C at angle π/3 and D at angle 5π/3. C = (3/2, 3√3/2), D = (3/2, −3√3/2). These share the same x-coordinate, so the distance CD = |3√3/2 − (−3√3/2)| = 3√3.
Quadrant I (0 to π/2): sin +, cos +, tan +
Quadrant II (π/2 to π): sin +, cos −, tan −
Quadrant III (π to 3π/2): sin −, cos −, tan +
Quadrant IV (3π/2 to 2π): sin −, cos +, tan −
Memory aid: "All Students Take Calculus" (All, Sine, Tangent, Cosine positive in Q I, II, III, IV respectively).
Coordinates on a circle of radius r: x = r cos θ, y = r sin θ
Trig ratios from coordinates: sin θ = y / r, cos θ = x / r, tan θ = y / x
Radius from coordinates: r = √(x² + y²)
Distance between two points on a circle: d = √((x₂ − x₁)² + (y₂ − y₁)²)
⚠️ When given a point like (−4.5, 1.25), the radius is not necessarily a whole number. You must compute r = √(x² + y²) before finding sin and cos. Do not assume r = 1.
⚠️ For angles beyond 2π or negative angles, first find the coterminal angle between 0 and 2π, then identify the quadrant and reference angle.
⚠️ When finding trig values from a line like y = 2x, the slope gives you tan θ directly (tan θ = 2). But sin and cos require you to build the right triangle with hypotenuse √5.
⚠️ The sign of each trig function depends on the quadrant. A common error is forgetting to apply the negative sign after computing with the reference angle.
⚠️ On the exam, "find the coordinates of point P" means give an exact answer with radicals, not a decimal approximation.
Q: An angle θ in standard position measures 4π/3 radians and intersects a circle of radius 6 at point P. What are the coordinates of P?
A: The reference angle is π/3. In quadrant III, both coordinates are negative. P = (6(−1/2), 6(−√3/2)) = (−3, −3√3).
Q: The terminal ray of angle θ passes through the point (−4.5, 1.25) on a circle centred at the origin. What is tan θ?
A: tan θ = y/x = 1.25/(−4.5) = −5/18.
Q: The terminal ray of θ lies along y = 2x in the first quadrant. What is sin θ?
A: Pick (1, 2). r = √5. sin θ = 2/√5 = 2√5/5.
Q: A point P = (√3/2, 3/2) lies on a circle at angle π/3. What is the radius?
A: r = √((√3/2)² + (3/2)²) = √(3/4 + 9/4) = √3.
Q: Points C and D lie on a circle of radius 3, at angles π/3 and 5π/3 respectively. What is the length of segment CD?
A: C = (3/2, 3√3/2), D = (3/2, −3√3/2). They share the same x-coordinate, so CD = 3√3/2 + 3√3/2 = 3√3.
unit circle, sine, cosine, tangent, standard position angle, terminal ray, reference angle, coordinates on a circle, radius, ASTC, quadrant signs, trig ratios from a point, y = mx terminal ray, precalculus unit 3, Topic 3.3, AP Precalculus