AP Precalculus 3.2 Sine, Cosine & Tangent — Radian Measures — Notes

📐 Part I — Sine, Cosine & Tangent

📍

Standard Position & Terminal Ray

How we place angles in the coordinate plane

Key Definitions

Standard Position: vertex at the origin, one ray on the positive x-axis.

Terminal Ray: the second ray — rotates away from the positive x-axis to form angle θ.

Point P(x, y): where the terminal ray intersects a circle of radius r. r = √(x² + y²), always positive.

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Defining Sin, Cos, and Tan

For any circle of radius r, with P = (x, y) on the circle

Sine, Cosine, and Tangent — Ratio Definitions (circle of radius r)

sin θ

sin θ = y / r

Ratio of vertical displacement of P from the x-axis to the distance from origin to P.

cos θ

cos θ = x / r

Ratio of horizontal displacement of P from the y-axis to the distance from origin to P.

tan θ

tan θ = y / x

The slope of the terminal ray. Also equals sin θ / cos θ.

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Memory aid

sin = y/r (y is vertical — think "up") · cos = x/r (x is horizontal) · tan = y/x = slope of the terminal ray. Note: r is always positive.

✏️

Computing Sin, Cos, Tan — Examples 1 & 2

Identify x, y, r then apply the ratios

📌 Example 1 — Circle of radius 3, P = (x, y). Which statement about cos θ is TRUE?

cos θ = x/3 — ratio of the horizontal displacement of P from the y-axis to the distance from origin to P (= radius = 3). This is the correct definition.

✗

B is wrong (that's sin = y/r). C is wrong (that's tan = y/x). D is wrong (that would be r/x).

Answer: (A) cos θ = x/3

📌 Example 2 — Circle of radius 5, P = (3, −4). Find sin θ, cos θ, tan θ.

x = 3, y = −4, r = 5. Verify: √(9+16) = √25 = 5 ✓

sin θ

−4/5

cos θ

3/5

tan θ

−4/3

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Reflections of Point P — Examples 3–6

P=(x,y) reflected across axes and origin

Point	Reflection of P = (x, y)	Coordinates	sin θ	cos θ	tan θ
P	Original	(x, y)	y/r	x/r	y/x
Q	Across y-axis (Ex 5)	(−x, y)	y/r	−x/r	−y/x
R	Across origin (Ex 6)	(−x, −y)	−y/r	−x/r	y/x
S	Across x-axis (Ex 4)	(x, −y)	−y/r	x/r	−y/x

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Key patterns

Ex 4 (S): sin = −y/r — x-axis reflection flips y → sin negates.
Ex 5 (Q): cos = −x/r — y-axis reflection flips x → cos negates.
Ex 6 (R): tan = (−y)/(−x) = y/x — origin reflection flips both → cancel → tan unchanged.

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Terminal Ray as a Line — Example 7

y = mx in QII — find x from x²+y²=r², then sin/cos/tan

📌 Example 7 — θ in Q2, terminal ray along y = −3x. Find sin θ, cos θ, tan θ.

y = −3x in Quadrant II (x < 0, y > 0)

Substitute y = −3x into x²+y²=r²: x² + 9x² = r² → 10x² = r² → x = ±r/√10.

Q2: x < 0 → x = −r/√10. Then y = −3(−r/√10) = 3r/√10.

Apply sin = y/r, cos = x/r. For tan: slope of the line = −3.

sin θ

3/√10

cos θ

−1/√10

tan θ

−3

= slope of line

🌟

tan θ = slope — always!

When the terminal ray has equation y = mx, then tan θ = m directly. No need to compute x and y separately — just read the slope. In Ex 7: slope = −3, so tan θ = −3.

📏 Part II — Radian Angle Measures

📐

What is a Radian?

Arc length divided by radius — a natural way to measure angles

Radian Measure

θ (in radians) = arc length / radius = s / r

One full revolution = 2π radians (circumference = 2πr, so 2πr/r = 2π).

For a unit circle (r = 1): θ numerically equals the arc length.

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Positive Angle

Counterclockwise rotation from positive x-axis. Standard direction.

↻

Negative Angle

Clockwise rotation from positive x-axis. Always negative value.

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Key radian facts

Full circle = 2π · Half = π · Quarter = π/2 · One-third = 2π/3
For fraction f of a circle: θ = f · 2π (positive if CCW, negative if CW)

✏️

Finding Radian Measures — Examples 1, 2 & 3

Arc length / radius · fraction of circle × 2π

📌 Example 1 — Quarter circle of radius 4, counterclockwise. Find θ in radians.

Arc length = (1/4) × 2π(4) = 2π

θ = arc / r = 2π / 4 = π/2

θ = π/2 radians

📌 Example 2 — One-third of a circle, radius 3, clockwise. Find θ.

Arc = (1/3) × 2π(3) = 2π · θ = 2π/3 in magnitude

Clockwise → negative → θ = −2π/3

θ = −2π/3 radians

📌 Example 3 — Find radian measure for each arc description.

a) 1/6 of a circle, CCW

θ = (1/6)(2π) = π/3

b) 2 full revolutions, CW

θ = 2(−2π) = −4π

c) 3/4 of a circle, CCW

θ = (3/4)(2π) = 3π/2

d) 7/8 of a circle, CCW

θ = (7/8)(2π) = 7π/4

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All 16 Standard Radian Angles — Example 4

Q1 values π/6, π/4, π/3, π/2 → Q2–Q4 by symmetry

All 16 Standard Radian Positions (counterclockwise from 0)

Quadrant I

π/6

π/4

π/3

π/2

Quadrant II

2π/3

3π/4

5π/6

Quadrant III

7π/6

5π/4

4π/3

3π/2

Quadrant IV

5π/3

7π/4

11π/6

2π

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Symmetry shortcut

Q2: π − (Q1 angle) e.g. π − π/6 = 5π/6 · Q3: π + (Q1 angle) e.g. π + π/6 = 7π/6 · Q4: 2π − (Q1 angle) e.g. 2π − π/6 = 11π/6

⭕

Sin, Cos, Tan on the Unit Circle — Examples 5 & 6

When r = 1: sin θ = y and cos θ = x directly from coordinates

On the Unit Circle (r = 1)

sin θ

= y

y-coordinate of P

cos θ

= x

x-coordinate of P

tan θ

= y/x

= sin θ / cos θ

📌 Example 5 — P = (1/2, √3/2) on the unit circle. Find sin θ, cos θ, tan θ.

sin θ

√3/2

cos θ

1/2

tan θ

√3

(√3/2)÷(1/2)

📌 Example 6 — R = P reflected over y-axis = (−1/2, √3/2). Angle α at R. Find sin α, cos α, tan α.

sin α

√3/2

cos α

−1/2

tan α

−√3

(√3/2)÷(−1/2)

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Quick Reference

🔑 Key Formulas

sin θ = y/r · cos θ = x/r · tan θ = y/x

Unit circle: sin=y · cos=x · tan=y/x

tan θ = slope of terminal ray

θ = s/r · Fraction f → θ = f(2π)

CCW = positive · CW = negative

⚠️ Common Mistakes

❌ Mixing up sin and cos

sin = y/r (vertical). cos = x/r (horizontal). Remember: Sin → y (both one syllable "up").

❌ Forgetting signs in Q2/Q3/Q4

x is negative in Q2, Q3 → cos negative. y is negative in Q3, Q4 → sin negative. Always check the quadrant!

❌ Clockwise = positive angle

Clockwise = NEGATIVE radians. Counterclockwise = positive. Always confirm direction before writing the angle.

❌ tan undefined where?

tan = y/x is undefined when x = 0 (at π/2, 3π/2). cos θ = 0 at those same angles.

🧠 Ready to Practice? Take the Quiz →

Sine, Cosine & Tangent+ Radian Measures

Memory aid

Key patterns

tan θ = slope — always!

Key radian facts

Symmetry shortcut

Sine, Cosine & Tangent
+ Radian Measures