Trig Functions & Graphs

A2.G.1Degrees / radians01/04

Convert between degrees and radians.

Step01of 04

Two units measure angles. Degrees split a full circle into 360 parts; radians use the arc length on a unit circle, so a full circle is $2 π$ .

π radians = 180°

That single conversion factor handles every unit change.

A2.G.1 — Convert between degrees and radians.

Step 01 of 04

Two units measure angles. Degrees split a full circle into 360 parts; radians use the arc length on a unit circle, so a full circle is $2 π$ .

π radians = 180°

That single conversion factor handles every unit change.

Step 02 of 04

Degrees → radians. Multiply by $π /180°$ .

60° \cdot \frac{π}{180°} = \frac{π}{3}

135° \cdot \frac{π}{180°} = \frac{3 π}{4}

Radians → degrees. Multiply by $180°/ π$ .

\frac{5 π}{6} \cdot \frac{180°}{π} = 150°

Step 03 of 04

Common angles to memorize in both units.

0°	→ $0$
30°	→ $π /6$
45°	→ $π /4$
60°	→ $π /3$
90°	→ $π /2$
180°	→ $π$
270°	→ $3 π /2$
360°	→ $2 π$

Step 04 of 04

Why radians? They make calculus clean: derivatives and Taylor series of sin/cos look right ONLY in radians. They also tie directly to arc length: an arc of $θ$ radians on a circle of radius $r$ has length $r θ$ .

Key insight

π = 180°

. Multiply by

π /180

to go to radians; by

180/ π

to go to degrees. Memorize the unit-circle angles in both.

A2.G.2 — The unit circle — key angles.

Step 01 of 04

The unit circle is a circle of radius 1 centered at the origin. For any angle $θ$ measured from the positive x-axis, the point on the circle has coordinates $(cos θ, sin θ)$ .

This single picture defines sin and cos for ALL angles — not just the acute angles in a right triangle.

Step 02 of 04

The four key first-quadrant angles. Memorize these and the rest come from symmetry.

$0$	→ $(1, 0)$
$π /6$ (30°)	→ $(\frac{3}{2}, \frac{1}{2})$
$π /4$ (45°)	→ $(\frac{2}{2}, \frac{2}{2})$
$π /3$ (60°)	→ $(\frac{1}{2}, \frac{3}{2})$
$π /2$ (90°)	→ $(0, 1)$

First coordinate is cos, second is sin.

Step 03 of 04

Reference angles + quadrant signs handle every other angle.

Quadrant I (0 to π/2)	→sin +, cos +
Quadrant II (π/2 to π)	→sin +, cos −
Quadrant III (π to 3π/2)	→sin −, cos −
Quadrant IV (3π/2 to 2π)	→sin −, cos +

Mnemonic: ASTC ("All Students Take Calculus") tells you which functions are positive in each quadrant — All / Sine / Tangent / Cosine.

Step 04 of 04

Worked examples.

$sin (2 π /3)$	→= $sin (π - 2 π /3) = sin (π /3) = 3 /2$ (QII, sin +)
$cos (5 π /4)$	→= $- cos (π /4) = - 2 /2$ (QIII, cos −)
$tan (7 π /6)$	→= $+ tan (π /6) = 3 /3$ (QIII, tan +)

Key insight

On the unit circle, point coordinates ARE

(cos θ, sin θ)

. Memorize Q1 (4 acute angles), use reference angle + ASTC sign rule for everything else.

A2.G.3 — Graph sine and cosine — amplitude, period, shifts.

Step 01 of 04

General sinusoid form.

y = A sin (B (x - C)) + D

$A$ — amplitude	→vertical stretch: peak height = $∣ A ∣$ from midline
$B$ — angular frequency	→period = $2 π /∣ B ∣$
$C$ — phase shift	→horizontal shift right by $C$
$D$ — vertical shift	→midline at $y = D$

Step 02 of 04

The parent functions.

Forest:

sin x

. Grey dashed:

cos x

. Both have amplitude 1, period

2 π

$cos$ is just $sin$ shifted left by $π /2$ .

Step 03 of 04

Worked example. Describe the graph of $y = 3 sin (2 x) + 1$ .

Amplitude	→3
Period	→ $2 π /2 = π$
Vertical shift	→+1 (midline at y = 1)
Range	→[-2, 4]

Range = midline ± amplitude → $1 - 3 = - 2$ to $1 + 3 = 4$ .

Step 04 of 04

Reading the period.

$sin (x)$	→period $2 π$
$sin (2 x)$	→period $π$
$sin (x /3)$	→period $6 π$
$sin (π x)$	→period $2$

Key insight

Four parameters in

A sin (B (x - C)) + D

: amplitude

∣ A ∣

, period

2 π /∣ B ∣

, phase shift

C

, midline

y = D

. Each transforms the parent

sin

graph in a predictable way.

A2.G.4 — Solve basic trigonometric equations.

Step 01 of 04

Find ALL solutions, not just one. Sin and cos are periodic, so a single equation usually has infinitely many solutions across all real numbers.

Procedure: find the principal solutions in $[0, 2 π)$ , then add multiples of the period to express all solutions.

Step 02 of 04

Worked example. Solve $sin θ = \frac{1}{2}$ for $θ \in [0, 2 π)$ .

From the unit circle, $sin = 1/2$ at the reference angle $π /6$ . Sine is positive in Q1 and Q2:

θ = \frac{π}{6} or θ = π - \frac{π}{6} = \frac{5 π}{6}

Step 03 of 04

All real solutions. Add $2 π k$ for any integer $k$ :

θ = \frac{π}{6} + 2 π k or θ = \frac{5 π}{6} + 2 π k

Each value of $k$ ( $\dots, - 1, 0, 1, 2, \dots$ ) gives a distinct solution.

Step 04 of 04

Watch for the period of the inside. If the equation involves $sin (2 θ)$ , that has period $π$ — solutions repeat every $π$ , not $2 π$ .

For tan: solutions repeat every $π$ (tan has period $π$ ).

$sin θ = c$	→two solutions in $[0, 2 π)$ (unless $c = \pm 1$ )
$cos θ = c$	→two solutions in $[0, 2 π)$
$tan θ = c$	→one solution per period of $π$
$∣ c ∣ > 1$ for sin/cos	→no solution

Key insight

Find principal solutions in one period using reference angle + sign chart. Express ALL solutions by adding multiples of the function's period:

2 π k

for sin/cos,

π k

for tan.

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A2.G.1 – A2.G.4 · CCSS F.TF.A.1 / F.TF.A.2 / F.TF.B.5 / F.TF.B.7·Full curriculum →