1. Basic Definitions
Trigonometry studies relationships between angles and sides in right and non‑right triangles.
Right triangle ratios (definitions)
For an angle θ in a right triangle:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent = sin θ / cos θ
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent = sin θ / cos θ
Reciprocal identities
cosec θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ
Pythagorean identity
sin²θ + cos²θ = 1
Compound angle (for reference)
sin(A±B)=sinA cosB ± cosA sinB
cos(A±B)=cosA cosB ∓ sinA sinB
tan(A±B)= (tanA ± tanB)/(1 ∓ tanA tanB)
cos(A±B)=cosA cosB ∓ sinA sinB
tan(A±B)= (tanA ± tanB)/(1 ∓ tanA tanB)
2. Special angles (common values)
θ: 0° | 30° | 45° | 60° | 90°
sinθ: 0 | 1/2 | √2/2 | √3/2 | 1
cosθ: 1 | √3/2 | √2/2 | 1/2 | 0
tanθ: 0 | √3/3 | 1 | √3 | —
sinθ: 0 | 1/2 | √2/2 | √3/2 | 1
cosθ: 1 | √3/2 | √2/2 | 1/2 | 0
tanθ: 0 | √3/3 | 1 | √3 | —
3. Examples — Right triangles
Example 1: In right ΔABC, angle A = 30°, hypotenuse = 10 cm. Find opposite side (BC).
Solution: opposite = hypotenuse × sin30° = 10 × 1/2 = 5 cm.
Solution: opposite = hypotenuse × sin30° = 10 × 1/2 = 5 cm.
Example 2: Right Δ with adjacent = 8 cm, angle = 37°. Find opposite and hypotenuse (use sin≈0.601, cos≈0.799).
Solution: opposite = adjacent × tanθ = 8 × 0.753 ≈ 6.02 cm (or use sin).
hypotenuse = adjacent / cosθ = 8 / 0.799 ≈ 10.01 cm.
Solution: opposite = adjacent × tanθ = 8 × 0.753 ≈ 6.02 cm (or use sin).
hypotenuse = adjacent / cosθ = 8 / 0.799 ≈ 10.01 cm.
Example 3 (Pythagoras check): If legs are 6 and 8, hypotenuse = √(6²+8²)=10.
4. Non‑right triangles (Sine & Cosine rules)
Sine rule: a / sinA = b / sinB = c / sinC
Cosine rule: c² = a² + b² − 2ab cosC
Cosine rule: c² = a² + b² − 2ab cosC
Example 4 (Sine rule): In ΔABC, A=30°, a=10, B=45°. Find b.
Using a/sinA = b/sinB ⇒ b = (sin45° × a)/sin30° = (√2/2 × 10) / (1/2) = 10√2 ≈ 14.14.
Using a/sinA = b/sinB ⇒ b = (sin45° × a)/sin30° = (√2/2 × 10) / (1/2) = 10√2 ≈ 14.14.
Example 5 (Cosine rule): 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≈6.24.
c² = 7²+5²−2×7×5×cos60° = 49+25−70×0.5 = 74−35 = 39 ⇒ c=√39≈6.24.
5. Useful tips & problem strategies
- Sketch the triangle and mark given sides/angles.
- Decide whether to use SOHCAHTOA (right triangle), sine rule, or cosine rule.
- Watch for ambiguous case in sine rule (two possible angles).
- Use exact values (√2, √3) when possible to avoid rounding errors.
6. Practice problems (with brief answers)
1) Right Δ: hypotenuse 13, one leg 5. Find other leg. Ans: 12 (Pythagoras)
2) Right Δ: angle=22.5°, hypotenuse=10. Find opposite. Ans: 10×sin22.5°≈3.83
3) ΔABC: a=9, b=7, C=120°. Find c. Ans: Cosine rule → c≈14.14
4) Δ with A=40°, a=8, B=70°. Find b. Ans: sine rule → b=(sin70×8)/sin40≈14.86
7. Formula summary
SOHCAHTOA, reciprocal identities, sin²+cos²=1, sine and cosine rules, compound-angle formulas