Precalculus · Self-Study Series

Trig
Foundations

20 core problems from the most frequently missed topics. Work at your own pace — explanations unlock when you need them.

Score 0 / 20 ✦ start
0 answered20 total
Unit Circle & Special Angles
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Memory Point

ALL STUDENTS TAKE CALCULUS

Quadrant I → All positive · II → Sin only · III → Tan only · IV → Cos only

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Memory Point

1 — √2/2 — √3/2

sin 30° = 1/2 · sin 45° = √2/2 · sin 60° = √3/2 · Think: "the numbers go UP as angle goes UP"

01
What is sin 30°?
Enter a fraction like 1/2 → type just the numerator if denominator is 2, or type the decimal.
Worked Example On the unit circle at 30°, the y-coordinate (height) = 1/2. So sin 30° = 1/2 = 0.5.
SOH = sin = Opposite / Hypotenuse
📖 Explanation sin 30° = 1/2. Remember the unit circle: at 30° the point is (√3/2, 1/2). sin = y-coordinate = 1/2 = 0.5.
02
What is cos 60°?
Worked Example cos 45° = √2/2 ≈ 0.707. Similarly, cos 60° uses the special triangle 30-60-90.
CAH = cos = Adjacent / Hypotenuse
📖 Explanation cos 60° = 1/2 = 0.5. The 30-60-90 triangle has sides 1, √3, 2. cos 60° = adjacent/hypotenuse = 1/2.
03
What is tan 45°?
Worked Example tan = sin/cos. At 45°: sin 45° = cos 45° = √2/2. So tan 45° = (√2/2) ÷ (√2/2) = 1.
TOA = tan = Opposite / Adjacent
📖 Explanation tan 45° = 1. Since sin 45° = cos 45°, their ratio is exactly 1. At 45° the right triangle is isosceles.
04
In which quadrant is sin positive but cos negative?
Type the Roman numeral: I, II, III, or IV
Worked Example sin = y-coordinate on unit circle. cos = x-coordinate. Quadrant II: x < 0, y > 0 → cos negative, sin positive.
ALL STUDENTS TAKE CALCULUS → Q2 = Sin only positive
📖 Explanation Answer: II. In Quadrant II, x is negative (cos < 0) and y is positive (sin > 0).
05
What is sin 90°?
Worked Example At 90° on the unit circle, the point is (0, 1). sin = y-coordinate = 1.
Unit Circle Point at 90° = (0, 1)
📖 Explanation sin 90° = 1. The top of the unit circle is the point (0, 1), and sin = y = 1.
Radians & Degree Conversions
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Memory Point

π rad = 180° (the golden rule)

Degrees → Radians: multiply by π/180 · Radians → Degrees: multiply by 180/π

06
Convert 180° to radians.
Type the answer as a multiple of pi, e.g. if answer is 2π type: 2pi · If answer is just π type: pi
Worked Example 90° × (π/180) = π/2. So 180° × (π/180) = π/1 = π.
Multiply by π/180 to go Degrees → Radians
📖 Explanation 180° × (π/180) = π. This is the fundamental identity: π radians = 180 degrees.
07
Convert π/6 radians to degrees.
Type just the number (no degree symbol)
Worked Example π/4 × (180/π) = 180/4 = 45°. Use the same idea for π/6.
Multiply by 180/π to go Radians → Degrees
📖 Explanation (π/6) × (180/π) = 180/6 = 30°. The π cancels out, leaving 180/6 = 30.
08
How many radians is a full circle (360°)?
Type: 2pi
Worked Example Half circle = π radians (180°). So full circle = 2 × π = 2π radians.
Full circle = 2π (two pies make a whole!)
📖 Explanation A full rotation = 2π radians. Think of circumference: C = 2πr. For unit circle (r=1), one full trip around = 2π.
Pythagorean Identities
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Memory Point

sin²θ + cos²θ = 1 (the big one)

Divide both sides by cos²θ → tan²θ + 1 = sec²θ · Divide by sin²θ → 1 + cot²θ = csc²θ

09
If sin θ = 3/5, what is cos θ? (Assume θ is in Quadrant I)
Type a fraction like 4/5
Worked Example sin²θ + cos²θ = 1 · If sin θ = 1/2: cos²θ = 1 − 1/4 = 3/4, so cos θ = √3/2.
sin²+cos²=1 → cos²=1−sin² → cos=√(1−sin²)
📖 Explanation cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25. So cos θ = 4/5. (3-4-5 right triangle!)
10
Simplify: 1 − sin²θ = ?
Type: cos2 (meaning cos squared theta)
Worked Example From sin²θ + cos²θ = 1, subtract sin²θ: cos²θ = 1 − sin²θ.
Rearrange the big identity: 1−sin²=cos² and 1−cos²=sin²
📖 Explanation 1 − sin²θ = cos²θ. This is just a rearrangement of the Pythagorean identity.
Graphing · Period · Amplitude
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Memory Point

y = A sin(Bx + C) + D

Amplitude = |A| · Period = 2π/B · Phase shift = −C/B · Vertical shift = D

11
What is the period of y = sin(x)?
Type: 2pi
Worked Example y = sin(2x) has period = 2π/2 = π. The coefficient in front of x divides 2π.
Period = 2π/B · If B=1, period = 2π
📖 Explanation Period = 2π. For y = sin(x), B = 1, so period = 2π/1 = 2π ≈ 6.28. The wave repeats every 2π units.
12
What is the amplitude of y = −3 cos(x)?
Type just a number (always positive!)
Worked Example y = 5 sin(x) has amplitude = |5| = 5. Amplitude is always the absolute value.
Amplitude = |A| (absolute value, always positive)
📖 Explanation Amplitude = |−3| = 3. The negative sign flips the graph (reflection), but amplitude = height = always positive.
13
What is the period of y = cos(2x)?
Type: pi (since π = pi)
Worked Example y = sin(4x): period = 2π/4 = π/2. Larger B → shorter period (faster wave).
Bigger B = shorter period = faster oscillation
📖 Explanation Period = 2π/B = 2π/2 = π. With B=2, the cosine wave completes one full cycle in π units instead of 2π.
Reference Angles & Negative Angles
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Memory Point

Reference angle = acute angle to x-axis

sin(−θ) = −sin(θ) ODD · cos(−θ) = cos(θ) EVEN · "Sine is ODD, Cosine is EVEN"

14
What is the reference angle of 150°?
Type just the number (no degree symbol)
Worked Example 200° is in Q3. Reference angle = 200° − 180° = 20°. Always find the acute angle to the nearest x-axis.
Q2: ref = 180° − θ · Q3: ref = θ − 180° · Q4: ref = 360° − θ
📖 Explanation 150° is in Q2. Reference angle = 180° − 150° = 30°. The angle to the nearest x-axis (the negative x-axis here) is 30°.
15
What is cos(−60°)?
Type a fraction: 1/2
Worked Example cos(−45°) = cos(45°) = √2/2 · cosine is even: cos(−θ) = cos(θ). Negative angle doesn't change cosine.
cos is EVEN: cos(-x) = cos(x) (no sign change)
📖 Explanation cos(−60°) = cos(60°) = 1/2. Cosine is an even function, so the negative sign disappears.
Reciprocal Trig Functions
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Memory Point

co-functions are reciprocals of each other

csc = 1/sin · sec = 1/cos · cot = 1/tan · "csc goes with sin, sec with cos — they SHARE the same letter"

16
If sin θ = 1/2, what is csc θ?
Worked Example If sin θ = 3/4, then csc θ = 1 ÷ (3/4) = 4/3. Just flip the fraction.
csc = 1/sin → FLIP the fraction
📖 Explanation csc θ = 1/sin θ = 1/(1/2) = 2. Flip 1/2 to get 2.
17
What is sec 60°?
Type a number like 2
Worked Example sec 45° = 1/cos 45° = 1/(√2/2) = 2/√2 = √2. For sec 60°: find cos 60° first.
sec 60° = 1/cos 60° = flip cos 60°
📖 Explanation cos 60° = 1/2 → sec 60° = 1/(1/2) = 2. Always find cos first, then take its reciprocal.
Solving Basic Trig Equations
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Memory Point

REFERENCE ANGLE + QUADRANT SIGNS = answer

Step 1: Ignore sign, find reference angle · Step 2: Use ASTC to decide which quadrants · Step 3: Write solutions

18
Solve for θ in [0°, 360°): sin θ = 1
Type just the number of degrees (no symbol)
Worked Example sin θ = 0: θ = 0° and 180°. The sin function equals 0 at 0° and 180° in [0°, 360°).
sin = 1 only at the TOP of the unit circle
📖 Explanation sin θ = 1 only at θ = 90°. On the unit circle, the maximum y-value of 1 occurs only at the top (90°).
19
Solve: cos θ = 0 for θ in [0°, 360°).
How many solutions are there?
Type just the number
Worked Example sin θ = 0 has 2 solutions: 0° and 180°. Think about where the graph crosses the x-axis.
cos = 0 at the LEFT and RIGHT sides of the unit circle
📖 Explanation There are 2 solutions: θ = 90° and θ = 270°. These are the two points where the x-coordinate on the unit circle is 0.
20
Final Challenge
What is the exact value of sin(5π/6)?
Type a fraction: e.g. 1/2
Worked Example sin(2π/3): reference angle = π − 2π/3 = π/3 (60°). sin(π/3) = √3/2. Q2 → sin positive → answer = √3/2.
5π/6 is in Q2 → ref angle = π − 5π/6 = π/6 (30°) → sin positive
📖 Explanation 5π/6 is in Q2. Reference angle = π − 5π/6 = π/6 = 30°. sin 30° = 1/2. Q2 → sin positive. Answer = 1/2.
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