Trigonometry
Compound angles, double angles, 3D problems
Compound and Double Angle Formulae
Compound angle formulae:
• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
Double angle formulae (let B = A):
• sin 2A = 2 sin A cos A
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
Example
Proving an Identity
Prove: sin 2x / (1 + cos 2x) = tan x
LHS = 2sin x cos x / (1 + 2cos²x − 1)
= 2sin x cos x / 2cos²x
= sin x / cos x
= tan x = RHS ✓
Example
3D Trigonometry
A flagpole stands on top of a building. From point P on the ground, the angle of elevation to the top of the building is 35° and to the top of the flagpole is 42°. If P is 50 m from the base:
Building height = 50 tan 35° ≈ 35.0 m
Total height = 50 tan 42° ≈ 45.0 m
Flagpole = 45.0 − 35.0 = 10.0 m
Note
Remember
In compound angle problems, expand and simplify. For 3D problems, identify the right-angled triangles. Common angles: sin 45° = cos 45° = √2/2, sin 30° = 1/2, cos 60° = 1/2.
Key Vocabulary
Compound angleAn angle expressed as the sum or difference of two angles
Double angleAn identity involving 2A (e.g. sin 2A = 2sin A cos A)
IdentityAn equation true for all values of the variable
Angle of elevationThe angle measured upward from horizontal
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Compound angle
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