Algebra 101 - Unit 16

Lesson 107 — Trigonometry Review & Summary

This lesson brings together the essential ideas from Unit 15 & Unit 16 on trigonometry. Use this page as a concise reference and quick study guide before assessments or applied problems.

Key Definitions & Concepts

• Unit circle: point at angle θ has coordinates (cos θ, sin θ).
• Right-triangle ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
• Radian–degree conversion: 180° = Ï€ radians → multiply degrees by Ï€/180 to convert to radians.
• Periodicity: sin/cos have period 2Ï€; tan has period Ï€.
• Reference angles: use acute reference angle + quadrant to find trig values/signs.

Essential Identities (Quick List)

• Pythagorean: sin²θ + cos²θ = 1
• Quotient / Reciprocal: tan θ = sin θ / cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
• Sum & Difference: sin(A±B)=sinA cosB±cosA sinB, cos(A±B)=cosA cosB∓sinA sinB
• Double-angle: sin2A=2sinA cosA, cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A
• Half-angle (use sign by quadrant): sin²(A/2)=(1−cosA)/2, cos²(A/2)=(1+cosA)/2
• Product-to-sum (handy for integrals/identities): sinA sinB = ½[cos(A−B) − cos(A+B)]
• Law of Sines: a/sinA = b/sinB = c/sinC
• Law of Cosines: a² = b² + c² − 2bc cosA

Problem-Solving Strategies

1. Label diagrams: Always draw and label triangles and known quantities (sides/angles).
2. Choose method: Right-triangle ratios if right triangle; otherwise decide Law of Sines or Cosines.
3. Use identities: Transform expressions to simpler forms (rewrite in sin/cos when helpful).
4. Watch periodicity: When finding all solutions, add the appropriate period (2π or π) and include quadrant alternatives.
5. Check ambiguous SSA: If you apply Law of Sines with SSA, check for 0, 1, or 2 possible triangles.

Common Mistakes to Avoid

• Forgetting that tan θ is undefined where cos θ = 0.
• Using degrees and radians inconsistently—always match the expected unit for calculators and answers.
• Not checking the domain after algebraic manipulations (e.g., squaring can introduce extraneous solutions).
• Missing the second solution in the unit circle when solving sin θ = k (both θ and π−θ in [0,Ï€]).

Worked Quick Examples

1) Convert and evaluate: Find cos(5Ï€/6).
5π/6 = 150° → coordinates (−√3/2, 1/2) → cos(5π/6) = −√3/2.

2) Use Law of Cosines: Sides a=7, b=5, C=60°. Find c.
c² = 7² + 5² − 2·7·5·cos60° = 49+25 −70·0.5 = 74 −35 = 39 → c = √39 ≈ 6.245.

3) Solve trig equation: Solve sin x = −½ on [0, 2π).
Reference angle = π/6. Sin negative in QIII & QIV → x = 7π/6, 11π/6.

Practice Problems (Review)

1. Evaluate: sin(3π/4), cos(330°), tan(π/6).
2. Solve for 0 ≤ x < 2π: 2 sin x − 1 = 0.
3. Using Law of Sines: A=40°, B=65°, a=12. Find b and c.
4. Prove identity: 1 + tan²θ = sec²θ.
5. Find the exact value of cos 15° using sum/difference identities.