Trigonometry Learning Path

Move from function graph intuition to symbolic identity transformations. Each lesson builds on the existing Geometry Spot material with richer visuals, calculators, and practice pathways.

4–6 learning hours 6 structured lessons Intermediate level

Lesson Cards

Follow the order or dive into the specific concept you need. Every card links to our expanded lessons with diagrams, worked examples, and FAQ.

Lesson 1

Graph Lab

Sine Function Graphs

From y = sin x to y = A sin(ωx + φ) + b with domain, range, symmetry, and transformations.

Interpret amplitude, period, and phase shift parameters
Sketch monotonic intervals and symmetry axes

Lesson 2

Cosine Function Graphs

Use even-function symmetry to connect cosine graphs to sine, including translations and vertical shifts.

Leverage even symmetry around the y-axis
Convert cosine graphs to sine using phase shifts

Lesson 3

Tangent Function Graphs

Control asymptotes and period π behavior for y = A tan(ωx + φ).

Locate vertical asymptotes and centers
Determine monotonic intervals across periods

Lesson 4

Induction (Reduction) Formulas

Transform any angle ±α + k·π/2 into an acute reference angle using name change and quadrant signs.

Use the odd/even rule to decide sine/cosine/tangent names
Map quadrants to sign charts quickly

Lesson 5

Sum & Difference Identities

Master sin(α ± β), cos(α ± β), and tan(α ± β) for evaluation and transformations.

Proof sketches using rotation concepts
Apply identities to solve angle addition problems

Lesson 6

Identity Transformations

Use double-angle, half-angle, product-to-sum, and triple-angle identities in complex derivations.

Derive sin 2α, cos 2α, and tan 2α formulas
Switch forms using product-to-sum

Practice Tracks

Reinforce each concept with focused practice sets. Every set mirrors the lesson order and includes detailed explanations.

Function Graphs

15 questions on domain, range, symmetry, and monotonic intervals for sine/cosine/tangent.

Induction (Reduction)

10 problems reinforcing odd/even naming and quadrant sign decisions.

Sum & Difference

8 targeted exercises on sin(α ± β) and cos(α ± β) applications.

Identity Transformations

12 comprehensive problems on double/half angle and product-to-sum changes.

Interactive Calculators

Pair the calculators with your notes to validate transformations, reduce angles, and automate identity conversions.

Function Parameter Calculator

Analyze y = A sin(ωx + φ) or cosine variants for period, range, extrema, and symmetry axes.

Induction (Reduction) Calculator

Automate ±α + k·π/2 decomposition, odd/even rule, and quadrant sign detection.

Sum & Difference Calculator

Compute sin(α ± β), cos(α ± β), or tan(α ± β) with symbolic steps.

Identity Transformation Calculator

Explore double-angle, half-angle, and product-to-sum identities interactively with worked solutions.

Next Steps

Continue to triangle solving or revisit geometry calculators for multi-topic practice.

Triangle Solving Path

Apply trigonometric identities directly to sine/cosine laws and area formulas.

All Geometry Calculators

Explore calculators for vectors, triangles, circles, and more advanced topics.