Trigonometry Flashcards

All 20 cards in this deck

What is Pythagoras' theorem?

$a^2 + b^2 = c^2$, where $c$ is the hypotenuse (the side opposite the right angle).

Which Pythagorean triples should you recognise?

3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 — and any multiple of these.
e.g. 6, 8, 10 and 10, 24, 26

What are the steps to find a missing side in a right-angled triangle using trigonometry?

e.g. find the side opposite a 30° angle when the hypotenuse is 12 cm

1. Label the sides O, A, H relative to the given angle: O = $x$, H = 12.
2. Choose the ratio (SOH CAH TOA): $\sin 30° = \frac{x}{12}$.
3. Rearrange and solve: $x = 12 \sin 30° = 6$ cm.

What are the steps to find a missing angle in a right-angled triangle using trigonometry?

e.g. opposite = 5 cm, hypotenuse = 10 cm

1. Label the sides: O = 5, H = 10.
2. Choose the ratio: $\sin \theta = \frac{5}{10}$.
3. Use the inverse function: $\theta = \sin^{-1}(0.5) = 30°$.

What are the side ratios of a 30°, 60°, 90° triangle?

$1 : \sqrt{3} : 2$ — the side opposite 30° is 1, opposite 60° is $\sqrt{3}$, and the hypotenuse is 2.

What are the side ratios of a 45°, 45°, 90° triangle?

$1 : 1 : \sqrt{2}$ — the two equal sides are opposite the 45° angles and the hypotenuse is $\sqrt{2}$ times a shorter side.

What are the steps to find the length of a chord from the radius and the perpendicular distance of the chord from the centre?

e.g. radius 5 cm, chord 3 cm from the centre

1. The perpendicular from the centre bisects the chord, giving a right-angled triangle with hypotenuse 5 and one side 3.
2. Pythagoras for half the chord: $\sqrt{5^2 - 3^2} = 4$.
3. Double it: chord $= 8$ cm.

What is the sine rule?

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ — use it when you know a side together with the angle opposite it.

True or false? Using the sine rule to find an angle always gives just one possible answer.

False. There is also an obtuse possibility, $180° - \theta$, which must be checked (the ambiguous case).

What is the cosine rule, in both its forms?

$a^2 = b^2 + c^2 - 2bc\cos A$ for a side; rearranged for an angle, $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.

What is the formula for the area of a triangle given two sides and the angle between them?

Area $= \frac{1}{2}ab\sin C$, where $C$ is the included angle between sides $a$ and $b$.

What are the steps to find a missing side from the area of a triangle?

e.g. area 24 cm², one side 12 cm, included angle 30°

1. Substitute into Area $= \frac{1}{2}ab\sin C$: $24 = \frac{1}{2} \times 12 \times b \times \sin 30°$.
2. Simplify the right-hand side: $24 = 3b$.
3. Divide: $b = 8$ cm.

When do you use the cosine rule rather than the sine rule?

When you know all 3 sides (to find an angle), or 2 sides and the included angle (to find the third side); otherwise use the sine rule.

What is the formula for the space diagonal of a cuboid with edges $a$, $b$ and $c$?

e.g. a cuboid 2 cm by 3 cm by 6 cm

$d = \sqrt{a^2 + b^2 + c^2}$
$d = \sqrt{4 + 9 + 36} = 7$ cm

What is meant by the angle between a line and a plane?

The angle between the line and its projection (its 'shadow') on the plane.

What are the steps to find the angle between a space diagonal of a cuboid and the base?

e.g. base diagonal 8 cm, vertical height 6 cm

1. Draw the right-angled triangle made by the base diagonal, the vertical edge and the space diagonal.
2. Use tan with opposite = height, adjacent = base diagonal: $\tan\theta = \frac{6}{8}$.
3. $\theta = \tan^{-1}(0.75) = 36.9°$ (1 d.p.).

What is meant by the angle between two planes?

The angle between two lines, one drawn in each plane, that both meet the line of intersection at right angles.

True or false? In a square-based pyramid, the angle between a sloping face and the base equals the angle between a sloping edge and the base.

False. The face angle is measured at the midpoint of a base edge using the slant height, so it is a different (larger) angle.

In a 3D problem, what do you do when the triangle you need has no right angle?

Sketch that triangle on its own with its true lengths, then use the sine rule or cosine rule.

True or false? In the formula $\text{area} = \frac{1}{2}ab\sin C$, $C$ can be any of the three angles of the triangle.

False. $C$ must be the angle between (included by) the two sides $a$ and $b$.