Trigonometry Mastery — Pre-Calculus

01

Trig Ratios — SOH-CAH-TOA

In a right triangle, the side opposite angle \(\theta\) is 8 and the hypotenuse is 17. What is \(\sin\theta\)?

💡SOH: Sin = Opposite / Hypotenuse CAH: Cos = Adjacent / Hyp TOA: Tan = Opp / Adj

02

Special Right Triangles

What is the exact value of \(\cos 45°\)?

💡45-45-90: sides = 1 : 1 : √2. cos45 = adj/hyp = 1/√2 = √2/2

03

Special Right Triangles

What is the exact value of \(\tan 60°\)?

💡30-60-90: sides = 1 : √3 : 2. tan60 = opp/adj = √3/1 = √3

04

Unit Circle — Coordinates

On the unit circle, what are the coordinates of the point corresponding to \(\theta = \dfrac{2\pi}{3}\)?

💡UNIT CIRCLE POINT: \((\cos\theta,\ \sin\theta)\). 2π/3 is in Q2 (x negative, y positive); reference angle = π/3

05

Reference Angles & Signs

In which quadrant is \(\sin\theta > 0\) but \(\cos\theta < 0\)?

💡ASTC / "All Students Take Calculus": Q1=All+, Q2=Sin+, Q3=Tan+, Q4=Cos+

06

Radian ↔ Degree Conversion

Convert \(210°\) to radians.

💡CONVERT: degrees × π/180 = radians. 180° = π rad

07

Pythagorean Identity

If \(\cos\theta = \dfrac{3}{5}\) and \(\theta\) is in Quadrant IV, find \(\sin\theta\).

💡PYTHAGOREAN ID: sin²θ + cos²θ = 1. In Q4: sinθ is NEGATIVE

08

Reciprocal Identities

Which expression is equivalent to \(\csc\theta\)?

💡RECIPROCALS: csc = 1/sin, sec = 1/cos, cot = 1/tan = cos/sin

09

Even / Odd Identities

What is the value of \(\sin(-30°)\)?

💡ODD/EVEN: sin(−θ) = −sin θ (ODD). cos(−θ) = cos θ (EVEN). tan(−θ) = −tan θ (ODD)

10

Double Angle Identity

If \(\sin\theta = \dfrac{1}{2}\), what is \(\sin 2\theta\)? (Assume \(\theta\) is in Quadrant I.)

💡DOUBLE ANGLE: sin 2θ = 2 sin θ cos θ. Find cos θ first using Pythagorean identity

11

Graphs — Amplitude

What is the amplitude of \(f(x) = -3\sin(2x)\)?

💡AMPLITUDE: |A| in y = A sin(Bx + C) + D. Negative A = reflection, but amplitude is always POSITIVE

12

Graphs — Period

What is the period of \(g(x) = \cos\!\left(\dfrac{x}{3}\right)\)?

💡PERIOD: T = 2π / |B| for sin/cos. Larger B → shorter period. Smaller B → longer period

13

Graphs — Phase Shift

What is the phase shift of \(h(x) = \sin\!\left(2x - \dfrac{\pi}{3}\right)\)?

💡PHASE SHIFT: −C/B in y = sin(Bx − C). Shift RIGHT if positive, LEFT if negative

14

Graphs — Vertical Shift

The function \(y = 2\cos(x) + 3\) has a maximum value of:

💡MAX/MIN: max = A + D, min = −A + D. D is the MIDLINE (vertical shift)

15

Inverse Trig — Domain & Range

What is \(\arcsin\!\left(\dfrac{\sqrt{2}}{2}\right)\)?

💡arcsin RANGE: [−π/2, π/2] only. arcsin(√2/2) asks "which angle in [−π/2, π/2] has sin = √2/2?"

16

Inverse Trig — Composition

Evaluate \(\cos\!\left(\arctan\!\left(\dfrac{3}{4}\right)\right)\).

💡DRAW A TRIANGLE: if tan θ = 3/4, then opp=3, adj=4, hyp=5 (Pythagorean triple). Then cos θ = adj/hyp

17

Solving Trig Equations

Find all solutions in \([0°, 360°)\) for \(2\sin\theta - 1 = 0\).

💡ISOLATE then REFERENCE: sinθ = 1/2 → reference angle = 30°. sin positive in Q1 and Q2

18

Solving with Identities

Solve for \(\theta \in [0, 2\pi)\): \(\cos^2\theta - \sin^2\theta = 1\).

💡DOUBLE ANGLE: cos²θ − sin²θ = cos 2θ. So cos 2θ = 1 → 2θ = 0, 2π, 4π → θ = 0, π

19

Sum Identity — Exact Value

Use a sum identity to find the exact value of \(\sin 75°\).

💡SUM IDENTITY: sin(A+B) = sinA cosB + cosA sinB. Write 75° = 45° + 30°

20

Law of Sines

In triangle \(ABC\), \(A = 40°\), \(B = 70°\), and side \(a = 10\). Find side \(b\) using the Law of Sines. (Round to 2 decimal places.)

💡LAW OF SINES: a/sinA = b/sinB = c/sinC. Pair a side with its OPPOSITE angle