This mini-lesson covers SL Geometry & Trigonometry: right-angled trig, the sine and cosine rules, the area = ½ab sin C formula, radians, arc length & sector area, the unit circle, exact values and identities.
Answer as you go and collect ⭐ stars. Press Start when ready.
Right-angled trig
SOHCAHTOA & Pythagoras
In a right-angled triangle, label sides relative to angle θ: opposite, adjacent, hypotenuse.
sin θ = opp/hyp · cos θ = adj/hyp · tan θ = opp/adjand Pythagoras: a² + b² = c²
Finding an angle: use the inverse, e.g. θ = tan⁻¹(opp/adj).
Quick check
Exact value
?What is the exact value of sin 30°?
Sine rule
The sine rule
For any triangle with sides a, b, c opposite angles A, B, C:
a/sin A = b/sin B = c/sin Cuse when you have a side and its opposite angle
Ambiguous case: when finding an angle with the sine rule, check whether an obtuse solution (180° − θ) also fits.
Calculate
Cosine rule
✎A triangle has b = 5, c = 7 and included angle A = 60°. Find side a to 2 decimal places.
Hint: a² = 5² + 7² − 2(5)(7)cos60° = 39, then a = √39.
Cosine rule & area
Cosine rule and area
Use the cosine rule when the sine rule cannot start (two sides + included angle, or three sides):
a² = b² + c² − 2bc cos Aarea of any triangle = ½·a·b·sin C
Worked example
b = 5, c = 7, A = 60°: a² = 25 + 49 − 2(5)(7)cos60° = 74 − 35 = 39, so a = √39 ≈ 6.24.
Quick check
Which ratio?
?In a right-angled triangle, cos θ equals:
Calculate
Triangle area
✎Find the area of the triangle with sides 5 and 7 and included angle 60°, to 2 d.p.
Hint: area = ½·5·7·sin60° = 17.5 × 0.866.
Radians
Radian measure
A radian is the angle subtended by an arc equal to the radius. Convert with π rad = 180°.
180° = π radso 90° = π/2, 45° = π/4, 60° = π/3
Sort it
Which tool solves it?
Tap a formula, then the situation it is for.
📐 Right-angled only
🔺 Any triangle
⭕ Circle (radians)
Circle formulae
Arc length & sector area
With θ in radians and radius r:
arc length s = rθ · sector area = ½ r²θthese simple forms only work in radians
Worked example
r = 6, θ = 1.2 rad: s = 6(1.2) = 7.2; area = ½(6²)(1.2) = 21.6.
Calculate
Arc length
✎A sector has radius r = 6 and angle θ = 1.2 radians. Find the arc length s.
Hint: s = rθ = 6 × 1.2.
Quick check
The identity
?Which statement is a correct trigonometric identity?
Unit circle
The unit circle & exact values
On the unit circle, a point at angle θ has coordinates (cos θ, sin θ). This defines sin and cos for all angles and gives the exact values:
✎For the same sector (r = 6, θ = 1.2 rad) find the sector area.
Hint: area = ½ r²θ = ½ × 36 × 1.2.
Match it
Match each expression to its exact value
Tap a statement on the left, then its matching partner on the right.
Expression
Exact value
Identities
Identities & equations
Two identities you must know:
sin²θ + cos²θ = 1 · tan θ = sin θ / cos θ
Solving trig equations: find all solutions in the given interval — the unit circle gives a second angle for each value.
Calculate
Find the angle
✎A right-angled triangle has opposite = 3 and adjacent = 4. Find θ in degrees to 2 d.p.
°
Hint: θ = tan⁻¹(3/4).
Quick check
Radians
?Which angle equals 45° in radians?
Strategy
Choosing the tool
Right-angled triangle → SOHCAHTOA. Non-right triangle → sine rule (side + opposite angle known) or cosine rule (SAS or SSS). Circle problems → work in radians.
Exam habit: keep exact surd/π values until the end and check your calculator is in the correct angle mode.