MCR3U Trigonometric Identities & Sine/Cosine Law: Grade 11 Functions Guide [Worked Proofs]

In Ontario MCR3U Grade 11 Functions, trigonometric identities are proven by algebraically manipulating the Left Side (LS) and Right Side (RS) independently using quotient identities (tan θ = sin θ / cos θ) and Pythagorean identities (sin²θ + cos²θ = 1). The ambiguous case of the Sine Law determines whether zero, one, or two valid triangles exist based on altitude h = b sin A.

Official Ontario Curriculum Breakdown: MCR3U

• Ministry Course Code: MCR3U (Functions, Grade 11, University Preparation)
• Curriculum Strand: Trigonometric Functions (Evaluating Special Angles, Verifying Identities, Non-Right Triangles)
• Fundamental Identities: Pythagorean (sin²θ + cos²θ = 1), Quotient (tan θ = sin θ / cos θ), Reciprocal (csc θ, sec θ, cot θ)
• Key Rule: Never move terms across the equals sign in an identity proof (LS = RS structure)
• Evaluation Structure: 70% Term Work, 30% Final Summative Exam / Evaluation

Worked Example 1: Proving a Rational Trigonometric Identity

Problem: Prove the following trigonometric identity for all valid values of θ: (1 - cos²θ) / (tan θ) = sin θ cos θ.

• Step 1 (Establish independent sides): Left Side (LS) = (1 - cos²θ) / (tan θ). Right Side (RS) = sin θ cos θ.
• Step 2 (Apply Pythagorean Identity to numerator): Recall sin²θ + cos²θ = 1, so 1 - cos²θ = sin²θ. LS becomes (sin²θ) / (tan θ).
• Step 3 (Apply Quotient Identity to denominator): Recall tan θ = sin θ / cos θ. LS becomes (sin²θ) / (sin θ / cos θ).
• Step 4 (Multiply by reciprocal): LS = sin²θ × (cos θ / sin θ).
• Step 5 (Cancel common factor sin θ): LS = sin θ cos θ.
• Step 6 (State conclusion): Since LS = sin θ cos θ and RS = sin θ cos θ, LS = RS. Therefore, the identity is proven.

Teacher's Strategy: Always convert tangent, cotangent, secant, and cosecant into sine and cosine as your very first step. Over 80% of Grade 11 identity proofs simplify once written in sines and cosines.

Worked Example 2: The Ambiguous Case of the Sine Law (SSA Triangle)

Problem: In triangle ABC, side a = 7 cm, side b = 10 cm, and angle A = 35°. Determine whether no triangle, one triangle, or two triangles can be formed, and find all possible angle measures for ∠B.

• Step 1 (Calculate triangle altitude h): h = b sin A = 10 sin(35°) ≈ 10(0.5736) = 5.74 cm.
• Step 2 (Compare side a to altitude h and side b): a = 7 cm. Notice h (5.74 cm) < a (7 cm) < b (10 cm).
• Step 3 (Deduce number of triangles): Because the opposite side a is longer than altitude h but shorter than adjacent side b, exactly TWO distinct triangles exist.
• Step 4 (Find acute angle B₁ using Sine Law): sin B / b = sin A / a => sin B = (10 sin 35°) / 7 = 5.736 / 7 ≈ 0.8194. B₁ = sin⁻¹(0.8194) ≈ 55.0°.
• Step 5 (Find obtuse angle B₂ using supplementary property): B₂ = 180° - 55.0° = 125.0°.
• Step 6 (Verify angle sums): Triangle 1 has ∠C₁ = 180° - 35° - 55° = 90°. Triangle 2 has ∠C₂ = 180° - 35° - 125° = 20°. Both are valid triangles.

Teacher's Strategy: If side a were strictly less than altitude h (a < 5.74 cm), the side would swing without reaching the base, resulting in zero triangles. Always sketch altitude h first.

The 5 Most Common Mistakes Ontario Students Make in MCR3U

• Treating identities like equations: Moving terms across the equal sign invalidates the proof on Ontario rubrics.
• Forgetting the second triangle in SSA problems: Assuming the calculator's acute inverse sine angle is the only solution.
• Miswriting Pythagorean variations: Writing 1 + cos²θ = sin²θ instead of 1 - cos²θ = sin²θ.
• Degree vs Radian mode errors: Entering degrees while the calculator is in radian mode during unit tests.
• Incomplete concluding statements: Failing to write 'LS = RS, therefore proven' costs communication points.

University Admissions & Prerequisite Cutoff Impact

MCR3U is the sole Grade 11 prerequisite for MHF4U Advanced Functions. Students dropping below 75% in Grade 11 trigonometry face severe hurdles in Grade 12 calculus and university admissions.

Diagnostic Readiness Check: Test Your Knowledge

• What is tan θ expressed in terms of reciprocal identities?: Answer: tan θ = 1 / cot θ or sin θ / cos θ.
• What is the exact value of sin(45°)?: Answer: 1 / √2 or √2 / 2.
• If in an SSA triangle side a < altitude h, how many triangles exist?: Answer: Zero triangles exist.