Master Unit 8 Fundamental Laws of Trigonometry quickly and score 90+ marks in FBISE exams.
hsanotes.com notes features:
- Complete solved Exercise 8.1, 8.2, 8.3 & Review Exercise 8
- Addition/subtraction formulas
- Double/half-angle identities
- Product-to-sum & sum-to-product
- Multiple angle proofs
- Identity verifications
- Review MCQs & questions
- Free PDF download
Target: FBISE students + top schools: Aitchison College, Beaconhouse, LGS, Roots Ivy, KIPS, Garrison, OPF, IMCB, F.G. Colleges.
Key Solved Questions (Exercise 8.1):
- (i) α=180°, β=60°: cos(α±β)=-1/2, sin(α+β)=-√3/2, sin(α-β)=√3/2, tan(α+β)=√3, tan(α-β)=-√3
(ii) α=90°, β=60°: cos(α+β)=-√3/2, cos(α-β)=√3/2, sin both=1/2, tan(α+β)=-1/√3, tan(α-β)=1/√3
(iii) α=180°, β=30°: cos both=-√3/2, sin(α+β)=-1/2, sin(α-β)=1/2, tan(α+β)=1/√3, tan(α-β)=-1/√3
(v) α=π/2, β=π/6: cos(α+β)=-1/2, cos(α-β)=1/2, sin both=√3/2, tan(α+β)=-√3, tan(α-β)=√3 - cos 15° = (√6 + √2)/4
cos 165° = -(√6 + √2)/4
cos 345° = (√6 + √2)/4
sin 75° = (√6 + √2)/4 - cos 75° = (√6 – √2)/4
cos 105° = -(√6 – √2)/4
cos 285° = (√6 – √2)/4
sin 15° = (√6 – √2)/4
Key Solved Questions (Exercise 8.2):
- cos θ = 3/5 (QI): sin 2θ = 24/25, cos 2θ = -7/25, tan 2θ = -24/7
Half: sin(θ/2) = 1/√5, cos(θ/2) = 2/√5, tan(θ/2) = 1/2 - tan θ = 12/5 (QIII): sin 2θ = -120/169, cos 2θ = -119/169, tan 2θ = 120/119
Half: sin(θ/2) = 3/√13, cos(θ/2) = -2/√13, tan(θ/2) = -3/2 - sin 2θ = 24/25 (QII): cos θ = -3/5, tan θ = -4/3
- sin 15° cos 15° = 1/4
- sin⁴α cos²α = (2 – cos 2α – 2 cos 4α + cos 6α)/32
- Many identities verified (e.g., sin 4θ = 4 sin θ cos θ (cos²θ – sin²θ), 2 cot θ sin²θ = sin 2θ)
Key Solved Questions (Exercise 8.3):
- sin A cos B = 1/2 [sin(A+B) + sin(A-B)]
cos A sin B = 1/2 [sin(A+B) – sin(A-B)]
cos A cos B = 1/2 [cos(A+B) + cos(A-B)]
sin A sin B = 1/2 [cos(A-B) – cos(A+B)] - 4 sin A cos B = 2 [sin(u) + sin(v)] where u=A+B, v=A-B
- sin 3θ + sin θ = 4 cos²θ sin θ
- sin 3α cos α = (sin 4α + sin 2α)/2
- 2 cos 2u cos u + sin 2u sin u = 2 cos³ u
- sin 20° sin 40° sin 80° = √3/8
Review Exercise 8 Highlights:
(i) sin(45°-30°) = sin45° cos30° – cos45° sin30° = (√6 – √2)/4
MCQs on addition formulas, exact values, identities.
View full PDF for complete solutions, diagrams & tables.
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