Precalc Test Review

4.1 Angles & Conversions

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#1-5 Fill-in-the-blank vocab

Just memorize these five:

1. Sum = 90° → complementary 2. Same terminal side → coterminal 3. Negative angles rotate clockwise 4. 180° = π radians 5. 1° = 60 minutes, 1 min = 60 sec, so 1° = 3600 sec

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#6 What quadrant is 200° in?

Quadrant III

Q1: 0°-90° Q2: 90°-180° Q3: 180°-270° ← 200° is here Q4: 270°-360°

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#7 What quadrant is 4π/5 in?

Quadrant II

Convert so you can think about it: (4π/5)(180/π) = 720/5 = 144° 144° is between 90° and 180° → Q2

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#8 Sketch 230° in standard position

Terminal side in Q3, 50° past the -x axis

1. Initial side = positive x-axis (always) 2. Rotate counterclockwise (positive angle) 3. 230° = 180° + 50° 4. So go past the -x axis by 50° → land in Q3 5. Draw arrow on terminal side, arc from initial to terminal

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#9 Positive coterminal angle for 170°

530°

Process: just add 360° 170° + 360° = 530°

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#10 Negative coterminal angle for θ = 3π/4

−5π/4

Process: subtract 2π 3π/4 − 2π = 3π/4 − 8π/4 = −5π/4

Your quiz: you did this in degrees instead of radians. Stay in whatever form they give you.

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#11 Convert 53.5812° → degrees, minutes, seconds

53° 34' 52.3"

1. Whole degrees = 53° 2. Take the decimal part: 0.5812 3. Multiply by 60: 0.5812 × 60 = 34.872 4. Whole minutes = 34' 5. Take new decimal: 0.872 6. Multiply by 60: 0.872 × 60 = 52.3"

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#12 Convert 79°47'38" → decimal degrees

79.7939°

Process: degrees + min/60 + sec/3600 79 + 47/60 + 38/3600 = 79 + 0.7833 + 0.01056 = 79.7939°

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#13 Convert 80° → radians

4π/9

degrees × π/180 = radians 80 × π/180 = 80π/180 reduce: both ÷ 20 = 4π/9

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#14 Convert 5π/6 → degrees

150°

radians × 180/π = degrees (5π/6) × (180/π) π cancels: 5 × 180/6 = 900/6 = 150°

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#15 Arc length problem (s = rθ)

s = rθ — θ MUST be radians

1. If θ is in degrees, convert first: θ × π/180 2. Plug into s = rθ 3. Units match the radius units

Most common mistake: forgetting to convert to radians before plugging in.

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#16 Area of sector: r = 12 in, θ = 60°

75.4 in²

A = ½ r² θ (θ in radians!) 1. Convert: 60° × π/180 = π/3 2. A = ½(12²)(π/3) 3. = ½(144)(π/3) 4. = 72 × π/3 = 24π 5. = 75.4

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#17 Bike tire: 20 in radius, 100 RPM → mph?

≈ 11.9 mph

v = rω (ω in rad per time) 1. ω = 100 rev/min × 2π = 200π rad/min 2. v = 20 in × 200π = 4000π in/min 3. → ft: 4000π ÷ 12 = 333.3π ft/min 4. → mi/hr: × 60 ÷ 5280 5. = 333.3π × 60 / 5280 ≈ 11.9 mph

You lost points on Quiz 4.1 here — make sure you know if they give you radius or diameter. If diameter, divide by 2 first.

4.2 Unit Circle & Trig Evaluation

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#1-4 Fill-in-the-blank vocab

Memorize these four:

1. csc has reciprocal relationship with sine 2. cot = cosine divided by sine 3. At rest = at its equilibrium 4. Overlaps when shifted horizontally = periodic

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#5 sin 4π = ?

1. 4π = 2 full rotations (2π + 2π) 2. Back at the start: point (1, 0) 3. sin = y-coordinate = 0

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#6 cos(7π/4) = ?

√2/2

1. 7π/4 = 315° 2. Reference angle = 45° 3. 315° is in Q4 4. cos is positive in Q4 5. cos 45° = √2/2 → stays positive 6. Answer: √2/2

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#7 tan 600° = ?

√3 (positive!)

1. Subtract 360°: 600° − 360° = 240° 2. 240° is in Q3 3. Reference angle: 240° − 180° = 60° 4. Point at 240°: (−1/2, −√3/2) 5. tan = y/x = (−√3/2) / (−1/2) 6. negative ÷ negative = positive √3

Tan is POSITIVE in Q3 (neg ÷ neg = pos). Your quiz had −√3 which was wrong.

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#8 sec(−45°) = ?

√2

1. −45° is same spot as 315° → Q4 2. cos(−45°) = cos(45°) = √2/2 3. sec = 1/cos = 1/(√2/2) = 2/√2 4. Rationalize: 2/√2 × √2/√2 = 2√2/2 = √2

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#9 csc(7π/3) = ?

2√3/3

1. 7π/3 − 2π = 7π/3 − 6π/3 = π/3 (= 60°) 2. sin(π/3) = √3/2 3. csc = 1/sin = 1/(√3/2) = 2/√3 4. Rationalize: 2/√3 × √3/√3 = 2√3/3

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#10 cot(−270°) = ?

1. −270° + 360° = 90° 2. Point at 90° = (0, 1) 3. cot = x/y = 0/1 = 0

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#11 Sketch 400° in standard position

Full rotation + 40° more → Q1

1. 400° − 360° = 40° 2. Draw full CCW loop, then 40° more 3. Terminal side in Q1 at 40°

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#12 Calculator: tan(6) = ?

−0.2910

1. No degree symbol → RADIAN mode 2. Type: tan(6) = −0.2910

No ° symbol = radians. With ° symbol = degree mode. Always check.

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#13 Calculator: csc(0.8) = ?

1.3940

1. Radian mode (no °) 2. No csc button — type: 1 / sin(0.8) 3. = 1.3940

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SKIP Harmonic motion (d = 2cos(3t))

NOT ON THIS TEST — it's on the next one

4.3 Right Triangles & Applications

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#1-3 Fill-in-the-blank vocab

1. csc = hyp / opposite 2. cot = adjacent / opposite 3. sec = 1 / cosine

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#4 Solve the triangle: side = 7, angle = 33°

x ≈ 10.78, y ≈ 12.84, t = 57°

Setup: right triangle, one angle 33°, side opp = 7 1. Missing angle: 90° − 33° = 57° 2. Find adj (x): tan 33° = opp/adj = 7/x x = 7/tan 33° = 7/0.6494 ≈ 10.78 3. Find hyp (y): sin 33° = opp/hyp = 7/y y = 7/sin 33° = 7/0.5446 ≈ 12.84

SOH CAH TOA — label opp/adj/hyp relative to YOUR angle first, then pick the right ratio.

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#5 sin θ = 3/5 → find all 6 trig functions

Process: find the missing side, then write all ratios

sin = opp/hyp → opp = 3, hyp = 5 adj = √(5² − 3²) = √(25 − 9) = 4 Now write them all: sin = 3/5 cos = 4/5 tan = 3/4 csc = 5/3 sec = 5/4 cot = 4/3

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#6 sec θ = 3 → find all 6 trig functions

Process: flip to get cos, find sides, write ratios

sec = 3 → cos = 1/3 → adj = 1, hyp = 3 opp = √(9 − 1) = 2√2 sin = 2√2/3 cos = 1/3 tan = 2√2/1 = 2√2 csc = 3/(2√2) = 3√2/4 sec = 3 cot = 1/(2√2) = √2/4

Rationalize: no √ on the bottom. Multiply top & bottom by the √.

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#7 Calculator: tan(62°44'12") = ?

1.9405

1. Convert DMS → decimal: 62 + 44/60 + 12/3600 = 62.7367° 2. DEGREE mode 3. tan(62.7367) = 1.9405

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#8 Calculator: sec(28°7'55") = ?

≈ 1.1339

1. Convert: 28 + 7/60 + 55/3600 = 28.1319° 2. sec = 1/cos 3. 1/cos(28.1319) = 1.1339

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#9 30-60-90 triangle: hypotenuse = 14

x = 7, y = 7√3

30-60-90 ratio: 1 : √3 : 2 1. Hyp = 14 → that's the "2" in the ratio 2. Short leg (opp 30°): 14 ÷ 2 = 7 3. Long leg (opp 60°): 7 × √3 = 7√3

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#10 Angle of elevation — balloon problem

Quiz example: 36.9°

Quiz: plane at 3000 ft altitude, 5000 ft from you 1. DRAW THE TRIANGLE 2. Height (altitude) = opposite = 3000 3. Distance from you = hypotenuse = 5000 4. sin θ = opp/hyp = 3000/5000 = 0.6 5. θ = sin−¹(0.6) = 36.9°

Test version is a BALLOON. Same process — draw triangle, figure out which side is which, pick the right trig ratio, use inverse to find the angle.

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#11 Verify: (1 + sec θ)(1 − sec θ) = −tan²θ

Work to write on paper:

Start with LEFT side: (1 + sec θ)(1 − sec θ) = 1 − sec²θ ← difference of squares Use identity: sec²θ = 1 + tan²θ = 1 − (1 + tan²θ) = 1 − 1 − tan²θ = −tan²θ ✓

NEW Inverse Unit Circle — Only New Questions on Test

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NEW Find all θ (0°–360°) where sin θ = √2/2

45° and 135°

1. What angle has sin = √2/2? Ref angle = 45° 2. Sin is positive in Q1 and Q2 3. Q1: 45° 4. Q2: 180° − 45° = 135°

There can be MORE THAN ONE answer! Check every quadrant where that trig function has the right sign.

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NEW Find θ (0°–180°) where tan θ = −√3

120°

1. Ref angle for √3 → 60° 2. Tan is negative in Q2 and Q4 3. 0°–180° → Q4 is excluded 4. Q2: 180° − 60° = 120° 5. Check: at 120° point is (−1/2, √3/2) tan = (√3/2)/(−1/2) = −√3 ✓

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NEW Same problems but in radians (0 ≤ θ ≤ π)

Same answers, radian form:

sin θ = √2/2 → π/4 and 3π/4 tan θ = −√3 → 2π/3 Process is identical — just give answer in radians

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METHOD How to do ANY inverse unit circle question

1. Find the reference angle (which basic angle gives that value?) 2. Determine which quadrants the function is +/− sin+: Q1, Q2 | cos+: Q1, Q4 | tan+: Q1, Q3 3. Check which quadrants are in the given range 4. Calculate angle for each valid quadrant: Q1: ref angle Q2: 180° − ref Q3: 180° + ref Q4: 360° − ref

REF Quick Reference

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Unit Circle (cos, sin) at key angles

0° / 0: (1, 0) 30° / π/6: (√3/2, 1/2) 45° / π/4: (√2/2, √2/2) 60° / π/3: (1/2, √3/2) 90° / π/2: (0, 1) Other quadrants = same numbers, flip signs: Q2: (−cos, +sin) | Q3: (−cos, −sin) | Q4: (+cos, −sin)

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Where is each function positive?

"All Students Take Calculus" Q1: All positive Q2: Sin (& csc) positive Q3: Tan (& cot) positive Q4: Cos (& sec) positive

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Reciprocal pairs + special triangles

sin ↔ csc | cos ↔ sec | tan ↔ cot tan = sin/cos | cot = cos/sin 45-45-90: 1 : 1 : √2 30-60-90: 1 : √3 : 2 (short : long : hyp)

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All the formulas

deg → rad: × π/180 rad → deg: × 180/π Arc length: s = rθ Sector area: A = ½r²θ Linear speed: v = rω sec²θ = 1 + tan²θ θ ALWAYS in radians for s, A, and v

PRECALC TEST