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

📍

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.

🔢

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 = yr

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

cos θ

cos θ = xr

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

tan θ

tan θ = yx

The slope of the terminal ray. Also equals sin θcos θ.

🎯

Memory aid

sin = yr (y is vertical — think "up") · cos = xr (x is horizontal) · tan = yx = 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, centre at origin, P = (x, y) on the circle. Which statement about cos θ is TRUE?

✓ (A)

cos θ = x3

cos = xr and r = 3 ✓ — horizontal distance over radius

(B)

cos θ = y3

That's sin θ = yr, not cos ✗

(C)

cos θ = yx

That's tan θ = yx ✗

(D)

cos θ = 3x

That's rx — the reciprocal (sec θ) ✗

Answer: (A) cos θ = x3 · cos = xr, and r = 3

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

1

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

sin θ

−45

cos θ

35

tan θ

−43

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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	Quadrant	sin θ	cos θ	tan θ
P	Original	(x, y)	Q I x > 0, y > 0	yr	xr	yx
Q	Across y-axis (Ex 5)	(−x, y)	Q II x < 0, y > 0	yr	−xr	−yx
R	Across origin (Ex 6)	(−x, −y)	Q III x < 0, y < 0	−yr	−xr	yx
S	Across x-axis (Ex 4)	(x, −y)	Q IV x > 0, y < 0	−yr	xr	−yx

💡

Key patterns

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

🔧

Terminal Ray as a Line — Example 7

y = mx in QII — find x from x²+y²=r², then sincos/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)

1

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

2

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

3

Apply sin = yr, cos = xr. 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 (slope) directly. No need to compute x and y separately — just read the slope. In Ex 7: slope = −3, so tan θ = −3.

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What is a Radian?

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

Radian Measure

θ (in radians) = arc lengthradius = sr

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

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

Positive Angle

Counterclockwise ↺ rotation from positive x-axis. θ > 0.

Negative Angle

Clockwise ↻ rotation from positive x-axis. θ < 0.

🔑

Key radian facts

Full circle = 2π · Half = π · Quarter = π2 · One-third = 2π3
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.

1

Arc length = 14 × 2π(4) = 2π

2

θ = arcr = 2π4 = π2

θ = π2 radians

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

1

Arc = 13 × 2π(3) = 2π · θ = 2π3 in magnitude

2

Clockwise → negative → θ = −2π3

θ = −2π3 radians

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

a) 16 of a circle, CCW

θ = 16(2π) = π3

b) 2 full revolutions, CW

θ = 2(−2π) = −4π

c) 34 of a circle, CCW

θ = 34(2π) = 3π2

d) 78 of a circle, CCW

θ = 78(2π) = 7π4

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All 16 Standard Radian Angles — Example 4

Q1 values π6, π4, π3, π2 → Q2–Q4 by symmetry

All 16 Standard Radian Positions (counterclockwise from 0)

Quadrant I

0

π6

π4

π3

π2

Quadrant II

2π3

3π4

5π6

π

Quadrant III

7π6

5π4

4π3

3π2

Quadrant IV

5π3

7π4

11π6

2π

🔄

Symmetry shortcut — from any Q1 angle

Start with your Q1 reference angle, then apply the formula for each quadrant:

Q2 → π − (Q1 angle) e.g. π − π6 = 5π6

💡 Numerator is 1 less than denominator. The numerator of the result = (denominator − 1).
denominator = 6 → numerator = 5 · denominator = 4 → numerator = 3 · denominator = 3 → numerator = 2

Q3 → π + (Q1 angle) e.g. π + π6 = 7π6

💡 Numerator is 1 higher than denominator. The numerator of the result = (denominator + 1).
denominator = 6 → numerator = 7 · denominator = 4 → numerator = 5 · denominator = 3 → numerator = 4

Q4 → 2π − (Q1 angle) e.g. 2π − π6 = 11π6

💡 Numerator is 1 less than twice the denominator. Numerator = (2 × denominator − 1).
denominator = 6 → numerator = 11 · denominator = 4 → numerator = 7 · denominator = 3 → numerator = 5

⭕

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

= yx

= sin θcos θ

📌 Example 5 — P = (12, √32) on the unit circle. Find sin θ, cos θ, tan θ.

sin θ

√32

cos θ

12

tan θ

√3

(√32)÷(12)

📌 Example 6 — R = P reflected over y-axis = (−12, √32). Angle α at R. Find sin α, cos α, tan α.

sin α

√32

cos α

−12

tan α

−√3

(√32)÷(−12)

⚡

Quick Reference

🔑 Key Formulas

sin θ = yr · cos θ = xr · tan θ = yx

Unit circle: sin=y · cos=x · tan=yx

tan θ = slope of terminal ray

θ = sr · Fraction f → θ = f(2π)

CCW = positive · CW = negative

⚠️ Common Mistakes

❌ Mixing up sin and cos

sin = yr (vertical). cos = xr (horizontal). Remember: Sin → y (both one syllable "up").

❌ Forgetting signs in Q2Q3/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 = yx is undefined when x = 0 (at π2, 3π2). cos θ = 0 at those same angles.

Sine, Cosine & Tangent+ Radian Measures

Memory aid

Key patterns

tan θ = slope — always!

Key radian facts

Symmetry shortcut — from any Q1 angle

🔗 Related Topics

Sine, Cosine & Tangent
+ Radian Measures