Source: Unit 6c Unit Circle Packet, Texas A&M University
Tags: sin, cos, tan, csc, sec, cot, trig functions unit circle, evaluating trig functions, reciprocal trig functions, trig values at special angles, undefined trig values
On the unit circle, sin θ is the y-coordinate, cos θ is the x-coordinate, and tan θ is y/x. The other three trig functions (csc, sec, cot) are simply reciprocals. Evaluating any trig function at a special angle means reading the right coordinate and possibly dividing.
sin θ
The y-coordinate of the point where the terminal ray meets the unit circle.
cos θ
The x-coordinate of that same point.
tan θ
The ratio y/x, equivalently sin θ / cos θ. Undefined when cos θ = 0 (at π/2 and 3π/2).
csc θ (cosecant)
The reciprocal of sine: 1/y. Undefined when sin θ = 0 (at 0, π, and 2π).
sec θ (secant)
The reciprocal of cosine: 1/x. Undefined when cos θ = 0 (at π/2 and 3π/2).
cot θ (cotangent)
The reciprocal of tangent: x/y. Undefined when sin θ = 0 (at 0, π, and 2π).
For any point P(x, y) on the unit circle at angle θ:
sin θ = y
cos θ = x
tan θ = y/x = sin θ / cos θ
csc θ = 1/y = 1/sin θ
sec θ = 1/x = 1/cos θ
cot θ = x/y = cos θ / sin θ
The coordinates of P can be written as (cos θ, sin θ). This is the single most important fact to internalise.
θ | cos θ (x) | sin θ (y) | tan θ (y/x) |
|---|---|---|---|
0 | 1 | 0 | 0 |
π/2 | 0 | 1 | undefined |
π | -1 | 0 | 0 |
3π/2 | 0 | -1 | undefined |
2π | 1 | 0 | 0 |
Tangent is undefined at π/2 and 3π/2 because the x-coordinate is 0 there, and division by zero is not defined.
θ | cos θ | sin θ | tan θ |
|---|---|---|---|
π/4 | √2/2 | √2/2 | 1 |
3π/4 | -√2/2 | √2/2 | -1 |
5π/4 | -√2/2 | -√2/2 | 1 |
7π/4 | √2/2 | -√2/2 | -1 |
Tangent at any π/4-family angle is always 1 or -1, because the x and y magnitudes are equal.
π/6 family:
θ | cos θ | sin θ | tan θ |
|---|---|---|---|
π/6 | √3/2 | 1/2 | 1/√3 = √3/3 |
5π/6 | -√3/2 | 1/2 | -1/√3 = -√3/3 |
7π/6 | -√3/2 | -1/2 | 1/√3 = √3/3 |
11π/6 | √3/2 | -1/2 | -1/√3 = -√3/3 |
π/3 family:
θ | cos θ | sin θ | tan θ |
|---|---|---|---|
π/3 | 1/2 | √3/2 | √3 |
2π/3 | -1/2 | √3/2 | -√3 |
4π/3 | -1/2 | -√3/2 | √3 |
5π/3 | 1/2 | -√3/2 | -√3 |
To evaluate csc, sec, or cot, just find sin, cos, or tan first, then take the reciprocal.
csc(3π/2): sin(3π/2) = -1, so csc(3π/2) = 1/(-1) = -1.
sec(11π/6): cos(11π/6) = √3/2, so sec(11π/6) = 2/√3 = 2√3/3.
cot(5π/4): tan(5π/4) = 1, so cot(5π/4) = 1.
When a point P is reflected over the y-axis to get point R, the x-coordinate changes sign but the y-coordinate stays the same.
If P = (1/2, √3/2), then R = (-1/2, √3/2).
For the angle α whose terminal ray hits R:
sin α = √3/2 (same y-coordinate)
cos α = -1/2 (x flips sign)
tan α = (√3/2) / (-1/2) = -√3 (sign flips)
This is why sin is positive in Quadrant II but cos and tan are negative there.
The process, step by step:
Identify which angle you have and find its reference angle.
Look up (or recall) the coordinates from the first-quadrant version.
Apply the correct signs for the quadrant.
Read off whichever coordinate or ratio the question asks for.
If it is a reciprocal function, flip the result.
The six trig functions on the unit circle:
sin θ = y, cos θ = x, tan θ = y/x
csc θ = 1/y, sec θ = 1/x, cot θ = x/y
Positive trig values by quadrant (ASTC):
Quadrant I: All six are positive
Quadrant II: sin and csc positive
Quadrant III: tan and cot positive
Quadrant IV: cos and sec positive
⚠️ The question "find the value of each trig function" is bread-and-butter for this unit. Speed comes from having the unit circle coordinates memorised cold.
⚠️ Don't forget to rationalise denominators when giving exact values. For example, 1/√3 should be written as √3/3.
⚠️ Undefined values are common traps. tan(π/2) is undefined, not 0 or infinity. csc(π) is undefined because sin(π) = 0.
⚠️ When the angle is negative (e.g. cos(-11π/6)), find its coterminal positive angle by adding 2π. -11π/6 + 2π = π/6, so cos(-11π/6) = cos(π/6) = √3/2.
⚠️ sin(2π) = 0, not 1. Full rotation returns to (1, 0), and sine is the y-coordinate.
Q: What is tan(π/3)?
A: √3. At π/3, the coordinates are (1/2, √3/2), so tan = (√3/2) / (1/2) = √3.
Q: What is cos 0?
A: 1. The point at 0 radians is (1, 0), and cosine is the x-coordinate.
Q: What is csc(3π/2)?
A: -1. sin(3π/2) = -1, so csc = 1/(-1) = -1.
Q: What is sin(2π)?
A: 0. The point at 2π is (1, 0), and sine is the y-coordinate.
Q: What is tan(7π/4)?
A: -1. The coordinates at 7π/4 are (√2/2, -√2/2), so tan = (-√2/2) / (√2/2) = -1.
Q: What is tan(3π/4)?
A: -1. Coordinates at 3π/4 are (-√2/2, √2/2), so tan = (√2/2) / (-√2/2) = -1.
Q: What is cos(-11π/6)?
A: √3/2. The coterminal positive angle is -11π/6 + 2π = π/6. cos(π/6) = √3/2.
Q: What is cot(5π/4)?
A: 1. At 5π/4, both coordinates are -√2/2, so tan = 1, and cot = 1/1 = 1.
Q: What is sec(11π/6)?
A: 2√3/3. cos(11π/6) = √3/2, so sec = 1 / (√3/2) = 2/√3 = 2√3/3.
Q: What is sin(7π/6)?
A: -1/2. The reference angle is π/6, so the magnitudes are (√3/2, 1/2). Quadrant III makes both negative, so sin = -1/2.
Q: What is tan(2π/3)?
A: -√3. Coordinates at 2π/3 are (-1/2, √3/2), so tan = (√3/2) / (-1/2) = -√3.
Q: What is csc(-3π/2)?
A: 1. -3π/2 + 2π = π/2. sin(π/2) = 1, so csc = 1.
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