Unit 3: Trigonometric and Polar Functions

Unit 3 is about periodic behavior. It covers radian measure, arc length and sector area, the unit circle, the graphs of sine, cosine, and tangent and their transformations, inverse trigonometric functions, trigonometric identities, and polar coordinates and polar graphs.

3.1 Radian Measure

A radian is defined by the radius itself. One radian is the central angle that cuts off an arc whose length equals the radius. Because the full circumference is 2πr, a full rotation is 2π radians, which means 360° = 2π radians and therefore 180° = π radians. A single degree is π/180 of a radian.

To convert, multiply by the right fraction. Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Work 225° into radians: 225 × π/180 = 225π/180. Both numbers share the factor 45, so 225π/180 = 5π/4. Check it back: 5π/4 means 5/4 of 180°, and 5/4 × 180 = 225. So 225° = 5π/4 radians.

Now go the other way. Convert 11π/6 radians to degrees: 11π/6 × 180/π. The π cancels, leaving 11 × 180/6 = 11 × 30 = 330°. Check it back: 330 × π/180 = 330π/180 = 11π/6 after dividing top and bottom by 30. The two directions undo each other exactly.

3.2 Arc Length and Sector Area

Two formulas run this section, and both require the angle in radians. For a circle of radius r and a central angle θ, the arc length is s = rθ, and the area of the sector is A = (1/2)r²θ. Both come from taking a fraction θ/(2π) of the full circumference 2πr or the full area πr².

Work an arc length. Take r = 6 and θ = 2π/3. Then s = rθ = 6 × 2π/3 = 12π/3 = 4π. A quick sense check: 2π/3 is one third of a full rotation, and one third of the circumference 2π(6) = 12π is 4π. The formula agrees.

Work a sector area with the same numbers. Take r = 6 and θ = 2π/3. Then A = (1/2)r²θ = (1/2)(36)(2π/3) = 18 × 2π/3 = 36π/3 = 12π. Check it the same way: one third of the full area π(6)² = 36π is 12π. Both formulas are just the radian fraction of the whole.

3.3 The Unit Circle

The unit circle is the circle of radius 1 centered at the origin. For an angle θ measured from the positive x-axis, the point on the circle is (cos θ, sin θ). That makes cosine the x-coordinate and sine the y-coordinate, and it turns the five first-quadrant angles below into the exact values every later topic assumes you know.

Every row satisfies sin²θ + cos²θ = 1, which is a good way to check your memory: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 for π/6, and (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1 for π/4. A memory aid: the sine values across the row read √0/2, √1/2, √2/2, √3/2, √4/2, and the cosine values read the same list in reverse.

Angles outside the first quadrant come from reference angles plus quadrant signs. The sine keeps its sign pattern + + − − across quadrants I through IV, and cosine keeps + − − +. So cos(11π/6) = cos(330°): the reference angle is π/6 and the fourth quadrant makes cosine positive, giving √3/2.

3.4 Sinusoidal Graphs: Amplitude, Period, Phase Shift, Midline

A sinusoidal function is any function that can be written y = A sin(B(x − C)) + D, and the same form works with cosine. Each letter controls one visible feature. The amplitude is |A|, the height of the wave above and below the midline. The period is 2π/|B|, the horizontal length of one full cycle. The phase shift is C, how far the graph slides horizontally, right when C is positive. The midline is the horizontal line y = D.

Work one example completely. For y = 3 sin(2(x − π/4)) + 1: A = 3, so the amplitude is 3. B = 2, so the period is 2π/2 = π, meaning the wave completes a full cycle every π units instead of every 2π. C = π/4, so the whole graph shifts right by π/4. D = 1, so the midline is y = 1 and the wave oscillates between y = −2 and y = 4.

3.5 The Tangent Function

The tangent function is defined by tan θ = sin θ / cos θ, which means it is undefined wherever cos θ = 0. Its period is π, half the period of sine and cosine, because shifting by π flips the signs of both sine and cosine and the two negatives cancel in the ratio. The graph has zeros at every multiple of π and vertical asymptotes at x = π/2 + nπ, and it increases across each interval between consecutive asymptotes.

3.6 Inverse Trigonometric Functions

Sine, cosine, and tangent repeat forever, so they have no true inverse until their domains are restricted to one monotonic piece. Each inverse trigonometric function undoes its partner only within that restricted range, and every calculator answer for an inverse trig function lands inside the range shown here. There are no exceptions to memorize beyond this table.

Evaluate three examples, verifying each one. For arcsin(−√3/2): we need the angle in [−π/2, π/2] whose sine is −√3/2, which is −π/3. Verify: sin(−π/3) = −sin(π/3) = −√3/2. For arccos(−1/2): we need the angle in [0, π] whose cosine is −1/2, which is 2π/3. Verify: cos(2π/3) = −cos(π/3) = −1/2. For arctan(1): we need the angle in (−π/2, π/2) whose tangent is 1, which is π/4. Verify: tan(π/4) = (√2/2)/(√2/2) = 1.

3.7 Trigonometric Identities

Identities are equations that hold for every angle in their domains, and the CED tests them as equivalent representations of the same expression. The table below gives the full set for this course, each checked numerically at θ = π/6 so you can see that both sides really do agree.

Two extra Pythagorean forms come from dividing sin²θ + cos²θ = 1 through by cos²θ or sin²θ: tan²θ + 1 = sec²θ, and 1 + cot²θ = csc²θ. They appear when a problem asks you to rewrite everything in terms of a single trig function.

3.8 Polar Coordinates

Polar coordinates locate a point by (r, θ) instead of (x, y). The number r is the directed distance from the pole (the origin), and θ is the angle from the polar axis (the positive x-axis). The conversion formulas come straight from right-triangle trigonometry: x = r cos θ, y = r sin θ, and in the other direction r² = x² + y² with tan θ = y/x.

Convert (r, θ) = (4, π/3) to rectangular form. Then x = 4 cos(π/3) = 4 × (1/2) = 2, and y = 4 sin(π/3) = 4 × (√3/2) = 2√3. Verify with r² = x² + y²: 2² + (2√3)² = 4 + 12 = 16 = 4². The conversion is consistent.

Convert (x, y) = (3, −3) to polar form with r ≥ 0. Then r² = 9 + 9 = 18, so r = √18 = 3√2, and tan θ = −3/3 = −1. The point is in quadrant IV, so θ = −π/4. Verify by converting back: 3√2 × cos(−π/4) = 3√2 × √2/2 = 3, and 3√2 × sin(−π/4) = −3. Both coordinates return exactly.

A negative r means walking the distance in the opposite direction of θ. The point (−4, π/3) sits at distance 4 along the angle π/3 + π = 4π/3, which is the rectangular point (−2, −2√3). Check: (−2)² + (−2√3)² = 4 + 12 = 16.

3.9 Polar Graphs

Polar equations r = f(θ) trace curves as θ sweeps around, and the CED focuses on three families. Circles: r = a is a circle of radius |a| centered at the pole, while r = 2a cos θ is a circle through the pole with diameter 2a along the polar axis. Limaçons have the form r = a ± b cos θ or r = a ± b sin θ: a cardioid when a = b, a dimpled loop when 1 < a/b < 2, an inner loop when a/b < 1, and a convex shape with no dimple when a/b ≥ 2. For example, r = 2 + 3 cos θ has a/b = 2/3, which is less than 1, so it has an inner loop.

Rose curves have the form r = a cos(nθ) or r = a sin(nθ), and the petal rule counts the petals: if n is odd there are n petals, and if n is even there are 2n petals. Each petal extends |a| from the pole. Verify the rule on two examples: r = 5 cos(3θ) has n = 3, which is odd, so 3 petals of length 5. And r = 4 sin(2θ) has n = 2, which is even, so 2 × 2 = 4 petals of length 4.

Confusions That Cost Points

Practice Questions

Original questions written for this guide in the style of the AP exam. Answers and explanations are on the next page, so complete the questions before checking them.

Answer Key

1. D. Multiply by π/180: 225 × π/180 = 225π/180 = 5π/4. A flips the conversion into 4π/5, which is 144°. B is 7π/4, which is 315°, a different quadrant entirely. C is 3π/4, which is 135°.

2. A. Arc length is s = rθ = 6 × 2π/3 = 12π/3 = 4π. B uses θ = π/3, half the given angle. C doubles the correct answer, a sign of multiplying by r twice. D uses θ = 2π, the full circle, instead of the given 2π/3.

3. C. The angle 11π/6 is 330°, in quadrant IV with reference angle π/6, and cosine is positive in quadrant IV, so cos(11π/6) = cos(π/6) = √3/2. A confuses cos(π/6) with sin(π/6), which is 1/2. B gets the magnitude right but the sign wrong; sine is negative in quadrant IV, cosine is not. D is sin(11π/6), the wrong coordinate.

4. B. The period is 2π/|B| = 2π/2 = π. A ignores the B value and reports the untransformed period of sine. C divides by B a second time. D folds the amplitude 3 into the period, but amplitude never affects horizontal length.

5. A. arcsin returns angles in [−π/2, π/2], and sin(−π/3) = −√3/2, so arcsin(−√3/2) = −π/3. B has the wrong magnitude: sin(−π/6) = −1/2. C has the wrong sign: sin(π/3) = √3/2, positive. D is outside the arcsin range entirely, and sin(5π/6) = 1/2 anyway.

6. C. r = √(2² + (2√3)²) = √(4 + 12) = 4, and tan θ = 2√3/2 = √3 with both coordinates positive, so θ = π/3. A keeps r = 2, confusing the radius with the x-coordinate. B uses 2π/3, the quadrant II supplement, but the point is in quadrant I. D uses π/6, the angle for the swapped point (2√3, 2).

7. D. For r = a cos(nθ) with n odd, the rose has n petals, and here n = 3. A has no basis in the rule and may come from halving. B confuses the petal length |a| = 5 with the petal count. C applies the even-n rule (2n) to an odd n.

8. B. The quotient identity gives tan θ = sin θ / cos θ. A is cot θ, the reciprocal of the correct answer. C is csc θ, the reciprocal of sine rather than the ratio of sine to cosine. D multiplies the two functions instead of dividing them.

One-Page Recall Check

Say each answer out loud before you look back, and mark the ones you cannot finish. Anything you cannot say out loud yet belongs in your flashcard deck. In Rycal, add those items to the Trigonometric and Polar Functions deck and let spaced review bring them back over the next few days.

• Define a radian and state the conversion factors between degrees and radians.
• Convert 225° to radians and 11π/6 radians to degrees from scratch, checking each result.
• State the arc length and sector area formulas and use them with r = 6 and θ = 2π/3.
• Write the unit-circle table for 0, π/6, π/4, π/3, and π/2, and explain how symmetry extends it to the other quadrants.
• For y = A sin(B(x − C)) + D, name the amplitude, period, phase shift, and midline in terms of A, B, C, and D.
• Describe the graph of y = tan x: its period, its zeros, and where its asymptotes are.
• State the domain and range of arcsin, arccos, and arctan.
• Evaluate arcsin(−√3/2), arccos(−1/2), and arctan(1), verifying each answer.
• State the Pythagorean, reciprocal, quotient, cofunction, even-odd, and periodicity identities.
• Convert between (r, θ) and (x, y) in both directions, and explain what a negative r means.
• Classify polar graphs as circles, limaçons, or rose curves, and state the petal rule.

Key terms for this unit

Radian, Degree-radian conversion, Arc length, Sector area, Unit circle, Reference angle, Amplitude, Period, Phase shift, Midline, Sinusoidal function, Tangent function, Inverse trigonometric function, arcsin, arccos, arctan, Pythagorean identity, Reciprocal identity, Quotient identity, Cofunction identity, Even-odd identity, Periodicity, Polar coordinates, Pole, Polar axis, Limaçon, Cardioid, Rose curve, Petal rule.

About this guide. Written for Rycal and aligned to the College Board AP Precalculus course framework, Unit 3. All questions and explanations are original Rycal writing. Rycal is independent and is not affiliated with or endorsed by the College Board.