Flashcards · 0580 · Topic 6

Trigonometry flashcards

33 cards on Topic 6, subtopics 6.1 to 6.6. Pick a subtopic or study them all.

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Pythagoras’ theorem: In a right-angled triangle: hypotenuse² = a² + b². The hypotenuse is the longest side, opposite the right angle. It is not on the List of formulas.
Pythagoras: finding the hypotenuse or finding a shorter side: Hypotenuse: square the two shorter sides, add, square root. Shorter side: square both, subtract (hypotenuse² − other side²), square root.
A 5 m ladder has its foot 1.4 m from a wall. How high up the wall does it reach?: √(5² − 1.4²) = √(25 − 1.96) = √23.04 = 4.8 m
Four Pythagorean triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
Is a triangle with sides 20, 21 and 29 right-angled?: Yes: 20² + 21² = 400 + 441 = 841 = 29². The right angle is opposite the 29 side.
The shorter sides of a right-angled triangle are 4 cm and 8 cm. Find the exact hypotenuse.Extended: √(16 + 64) = √80 = 4√5 cm

SOHCAHTOA: sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent
How do you label the sides before using sin, cos or tan?: Find the hypotenuse first (opposite the right angle). Then label the opposite and adjacent sides relative to the marked angle: they swap if you use the other angle.
The hypotenuse is 12 cm and the angle is 40°. Find the side opposite the angle.: opposite = 12 × sin 40° = 7.71 cm (3 s.f.)
The opposite side is 5 cm and the hypotenuse is 10 cm. Find the angle.: sin θ = 5/10 = 0.5, so θ = sin⁻¹(0.5) = 30°.
Angles of elevation and depressionExtended: Both are measured from the horizontal: elevation looking up, depression looking down. The angle of depression from A to B equals the angle of elevation from B to A (alternate angles).
Shortest distance from a point to a lineExtended: The perpendicular distance: measured along the line through the point at right angles to the given line.

Exact values of sin 0°, sin 30°, sin 45°, sin 60°, sin 90°Extended: 0, 1/2, √2/2, √3/2, 1
Exact values of cos 0°, cos 30°, cos 45°, cos 60°, cos 90°Extended: 1, √3/2, √2/2, 1/2, 0 (the sin values in reverse order).
Exact values of tan 0°, tan 30°, tan 45°, tan 60°Extended: 0, √3/3 (= 1/√3), 1, √3
Find the exact value of sin 60° × cos 30°Extended: (√3/2) × (√3/2) = 3/4
Exam trap with exact valuesExtended: Giving a rounded decimal such as 0.707 or 1.73 instead of √2/2 or √3. (0.5 for sin 30° is fine: it is exactly 1/2.)

Key values of y = sin x and y = cos x for 0° ≤ x ≤ 360°Extended: sin x: 0 at 0°, 1 at 90°, 0 at 180°, −1 at 270°, 0 at 360°. cos x: 1 at 0°, 0 at 90°, −1 at 180°, 0 at 270°, 1 at 360°. Both lie between −1 and 1.
Shape of y = tan x for 0° ≤ x ≤ 360°Extended: Repeats every 180°. 0 at 0°, 180° and 360°. Undefined at 90° and 270°, where it has vertical asymptotes.
Solve sin x = 0.5 for 0° ≤ x ≤ 360°Extended: x = 30° and x = 180° − 30° = 150°.
Solve cos x = 0.5 for 0° ≤ x ≤ 360°Extended: x = 60° and x = 360° − 60° = 300°. (For cosine use 360° − x, not 180° − x.)
Solve tan x = 1 for 0° ≤ x ≤ 360°Extended: x = 45° and x = 45° + 180° = 225°.

The sine rule, and when to use itExtended: a/sin A = b/sin B = c/sin C Use it when you know an angle and the side opposite it, plus one more side or angle.
The cosine rule, and when to use itExtended: a² = b² + c² − 2bc cos A Use it for two sides and the included angle, or for three sides (to find an angle).
The cosine rule rearranged to find an angleExtended: cos A = (b² + c² − a²)/(2bc) A negative value means angle A is obtuse.
Area of a triangle using sine. Find the area when a = 6 cm, b = 8 cm and C = 30°.Extended: Area = ½ab sin C, with C the angle between sides a and b. ½ × 6 × 8 × sin 30° = 12 cm²
AB = 7 cm, AC = 10 cm, angle BAC = 110°. Find BC.Extended: BC² = 49 + 100 − 140 cos 110° = 196.9… BC = 14.0 cm (3 s.f.). cos 110° is negative, so the last term adds.
The ambiguous case of the sine ruleExtended: When finding an angle, sin θ = k has two solutions between 0° and 180°: θ and 180° − θ. Check whether the obtuse one also fits the triangle.

How do you solve a 3D Pythagoras or trigonometry problem?Extended: Find the right-angled triangle inside the solid (often using a diagonal of the base or a face), sketch it flat, then use Pythagoras or SOHCAHTOA.
Space diagonal of a cuboid with edges a, b and cExtended: d = √(a² + b² + c²) For 6 × 8 × 10: √200 = 14.1 cm (3 s.f.).
Angle between a line and a planeExtended: The angle between the line and its projection on the plane (the line in the plane directly "below" it), found from a right-angled triangle.
A cuboid has base 6 cm by 8 cm and height 10 cm. Find the angle between the space diagonal and the base.Extended: Base diagonal = √(36 + 64) = 10 cm. tan θ = 10/10 = 1, so θ = 45.0°.
Find the exact length of the space diagonal of a cube of side 4 cmExtended: √(4² + 4² + 4²) = √48 = 4√3 cm (≈ 6.93 cm). Keep exact values until the final step.