Trigonometry: triangles, radians and graphs flashcards

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Radian
The angle at the centre of a circle subtended by an arc equal in length to the radius. π rad = 180°, so 1 rad ≈ 57.3°.
Bearing
A direction measured clockwise from north, written with three figures (e.g. 072°).
Back bearing
The bearing of A from B, found from the bearing of B from A by adding or subtracting 180°.
Segment of a circle
The region between a chord and the arc it cuts off.
Sector of a circle
The region between two radii and the arc joining their ends.
Included angle
The angle between two given sides of a triangle — the angle needed in ½ab sin C and in the cosine rule for a side.
Period of a function
The length of the smallest interval after which the graph repeats: 2π for sin and cos, π for tan.
Sine rule (two forms)
a/sin A = b/sin B = c/sin C (for a side); sin A/a = sin B/b = sin C/c (for an angle). (No — learn it)
Cosine rule for a side
a² = b² + c² − 2bc cos A. (Yes)
Cosine rule for an angle
cos A = (b² + c² − a²)/(2bc). (Yes (as a² = b² + c² − 2bc cos A; rearrange))
Area of a triangle from two sides and the included angle
Area = ½ab sin C. (No — learn it)
Arc length (θ in radians)
s = rθ. (No — learn it)
Sector area (θ in radians)
A = ½r²θ. (No — learn it)
Segment area (θ in radians)
½r²(θ − sin θ) = sector − triangle. (No — derive it)
Degrees ↔ radians
radians = degrees × π/180; degrees = radians × 180/π. (No — learn it)
Key features of y = sin x on 0 ≤ x ≤ 2π
Starts at (0, 0); max (π/2, 1); crosses at (π, 0); min (3π/2, −1); ends at (2π, 0). Period 2π.
Key features of y = cos x on 0 ≤ x ≤ 2π
Starts at max (0, 1); crosses at π/2 and 3π/2; min (π, −1); ends at (2π, 1). Symmetric about x = π.
Key features of y = tan x on 0 ≤ x ≤ 2π
Through (0, 0), (π, 0), (2π, 0); asymptotes x = π/2 and x = 3π/2; period π; no max or min.
y = a sin x compared with y = sin x
Stretch parallel to the y-axis, scale factor a: range −a ≤ y ≤ a (a > 0); zeros and period unchanged.
y = sin(x + c) compared with y = sin x
Translation c to the LEFT (vector (−c, 0)); shape and period unchanged.
y = sin bx compared with y = sin x
Stretch parallel to the x-axis, scale factor 1/b: period 2π/b.
Why can the sine rule give two triangles?
sin θ = sin(180° − θ), so an angle found from its sine may be acute or obtuse. Both are valid if the obtuse one still fits the angle sum.
When is there no ambiguity in finding an angle?
With the cosine rule (cos is negative for obtuse angles), or when the side opposite the given angle is the longer given side.
Why must θ be in radians in s = rθ?
The formula comes from the fraction θ/2π of the circumference 2πr, which uses 2π for a full turn.
Using the symmetry of y = sin x
sin(π − x) = sin x, so if α is a solution of sin x = k, then π − α is another (for k between −1 and 1).
Using the symmetry of y = cos x
cos(2π − x) = cos x, so if α is a solution of cos x = k, then 2π − α is another.
Triangle: A = 50°, B = 70°, a = 10 cm. Find b.
b = 10 sin 70°/sin 50° = 12.3 cm.
Sides 6 cm and 9 cm with 40° between them. Third side?
c² = 36 + 81 − 108 cos 40° → c = 5.85 cm.
Triangle with sides 4, 5 and 7. Largest angle?
cos θ = (16 + 25 − 49)/40 = −0.2 → θ = 101.5°.
Area of a triangle with sides 8 cm and 11 cm and an included angle of 30°?
½ × 8 × 11 × sin 30° = 22 cm².
Sector: r = 12 cm, θ = 0.75 rad. Arc length and area?
Arc = 12 × 0.75 = 9 cm; area = ½ × 144 × 0.75 = 54 cm².
Segment: r = 5 cm, θ = π/3. Area?
½ × 25 × (π/3 − √3/2) = 2.26 cm².
Convert 150° to radians and 1.5 rad to degrees.
150° = 5π/6; 1.5 rad = 85.9°.
Coordinates of the maximum of y = sin(x + π/4), 0 ≤ x ≤ 2π?
x + π/4 = π/2 → (π/4, 1).
Calculator mode
Degrees for angles given in degrees and for bearings; radians for arcs, sectors, segments and graphs on 0 ≤ x ≤ 2π — including inside the cosine rule.
Using ½ab sin C with the wrong angle
C must be the angle between sides a and b; find it first if necessary.
Perimeter of a sector
Arc + BOTH radii: 2r + rθ.
Rounding in multi-step triangle problems
Store full calculator values; round only final answers.
“Show that θ = 1.591” style questions
Write the value to more figures (e.g. 1.5912…) before rounding to the given answer.

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