A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Functions
CSCA Mathematics · Lesson 7 of 16Trigonometric graphs and triangles
Courbes trigonométriques et triangles · منحنيات الدوال المثلثية والمثلثات
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Sine and cosine repeat every 2π. Their graphs, a few identities and two triangle laws answer most CSCA trigonometry questions.
Key ideas
- y = A sin(ωx + φ): amplitude |A|, period 2π/|ω|.
- sin x is odd, cos x is even; tan x has period π.
- Law of sines: a/sin A = b/sin B = c/sin C = 2R.
- Law of cosines: a² = b² + c² − 2bc cos A.
Formulas to know
- sin(α ± β) = sin α cos β ± cos α sin β
- cos 2α = cos²α − sin²α = 1 − 2sin²α
- Triangle area = ½ bc sin A
Worked example
In triangle ABC, b = 3, c = 4 and A = 60°. Find a.
- Law: a² = b² + c² − 2bc cos A
- Replace: a² = 9 + 16 − 2·3·4·½ = 13
- Root: a = √13
Answer: a = √13 ≈ 3.61
Practice questions
1. Period of y = 3 sin(2x)?
2π · π ✓ · 3 · π/2
2π/2 = π.
2. Maximum value of y = 2 sin x + 1?
1 · 2 · 3 ✓ · 4
sin x ≤ 1, so y ≤ 2 + 1 = 3.
3. Area of a triangle with b = 4, c = 5 and A = 30°?
5 ✓ · 10 · 20 · 2.5
½ · 4 · 5 · ½ = 5.