Trigonometry flashcards
14 cards on Trigonometry: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 14
Trigonometry
Labelling the sides
In a right-angled triangle, how do you label the hypotenuse, the opposite and the adjacent?
Answer it in your head, then turn the card over.
Labelling the sides
Key points
- Mark the angle you are using first.
- Hypotenuse: the longest side, facing the right angle.
- Opposite: the side facing the angle you are using.
- Adjacent: the side left over, next to your angle.
Watch out
Label from your angle, never the right angle: from the other angle, opposite and adjacent swap over.
Trigonometry
SOH CAH TOA
What are the three trig ratios, and how do you choose which one to use?
Answer it in your head, then turn the card over.
SOH CAH TOA
Key points
- Find the two sides involved: the one you know and the one you want.
- Opposite and hypotenuse: sin. Adjacent and hypotenuse: cos. Opposite and adjacent: tan.
- SOH CAH TOA holds all three: Sin Opposite Hypotenuse, and so on.
Formulae
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
Choosing the ratio
You know the hypotenuse and want the adjacent.
A and H come together in CAH, so use cos.
Watch out
Label all three sides before choosing: guessing is how sin and cos get swapped.
Trigonometry
Finding a side: unknown on top
How do you find a missing side when it is on the top of the trig fraction?
Answer it in your head, then turn the card over.
Finding a side: unknown on top
Key points
- Label the sides and choose the ratio.
- Write the equation, such as sin 40° = .
- The unknown is on top, so multiply: x = 15 × sin 40°.
- Round only at the end, to the accuracy asked for.
Find x: hypotenuse 15 cm, angle 40°
O and H, so sin: sin 40° =
x = 15 × sin 40° = 9.64 cm (2 d.p.)
Watch out
Check the calculator is in degrees: sin 30 should give 0.5, and −0.988 means it is in radians.
Trigonometry
Finding a side: unknown on the bottom
How do you find a missing side when it is on the bottom of the trig fraction?
Answer it in your head, then turn the card over.
Finding a side: unknown on the bottom
Key points
- When the unknown is on the bottom, divide the known side by the trig value.
- cos 40° = becomes x = 8 ÷ cos 40°.
- Sense-check: the hypotenuse must be the longest side.
Find x: adjacent 8 cm, angle 40°
A and H, so cos: cos 40° =
x = 8 ÷ cos 40° = 10.4 cm (1 d.p.)
10.4 is longer than 8, as a hypotenuse should be.
Watch out
Multiplying instead gives 8 × cos 40° = 6.1, shorter than a leg, so it cannot be the hypotenuse.
Trigonometry
Finding a missing angle
How do you find a missing angle in a right-angled triangle?
Answer it in your head, then turn the card over.
Finding a missing angle
Key points
- Label the sides and choose the ratio from the two sides you know.
- Write the ratio as a fraction, such as tan θ = .
- Use the inverse function: θ = tan⁻¹. On most calculators, SHIFT then tan.
- Check the calculator is in degrees.
Opposite 9 cm, adjacent 4 cm
O and A, so tan: tan θ =
θ = tan⁻¹ = 66.0° (1 d.p.)
Watch out
An angle needs the inverse: sin 0.5 is not an angle, but sin⁻¹ 0.5 = 30°.
Trigonometry
The exact values
What are the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°?
Answer it in your head, then turn the card over.
The exact values
Key points
- They come up on the non-calculator paper, so learn them or rebuild them from two triangles.
- The cos list is the sin list backwards.
- tan θ = sin θ ÷ cos θ, so tan 45° = 1. tan 90° has no value.
Formulae
sin of 0°, 30°, 45°, 60°, 90°: 0, , , , 1
cos of 0°, 30°, 45°, 60°, 90°: 1, , , , 0
tan of 0°, 30°, 45°, 60°: 0, , 1,
Watch out
sin 30° and cos 60° are both , but sin 60° is : do not swap the 30° and 60° values.
Trigonometry
The 45° triangle
How do you work out sin 45°, cos 45° and tan 45° without a calculator?
Answer it in your head, then turn the card over.
The 45° triangle
Key points
- Draw a right-angled isosceles triangle with two sides of 1.
- By Pythagoras, its hypotenuse is .
- sin 45° = cos 45° = , which is .
- tan 45° = 1 ÷ 1 = 1, because the two shorter sides are equal.
Example
Hypotenuse² = 1² + 1² = 2, so the hypotenuse is .
Watch out
and are the same value, written two ways: either is exact.
Trigonometry
The 30° and 60° triangle
Which triangle gives the exact values for 30° and 60°, and what are they?
Answer it in your head, then turn the card over.
The 30° and 60° triangle
Key points
- Cut an equilateral triangle with sides of 2 in half: sides 1, and 2.
- The 30° is at the top, opposite the side of 1; the 60° is at the base.
- sin 30° = , cos 30° = , tan 30° = .
- sin 60° = , cos 60° = , tan 60° = .
Example
Height² = 2² − 1² = 3, so the height is .
Watch out
The short side of 1 faces the 30° angle, not the 60° one: sketch the triangle to check.
Trigonometry
HigherThe sine rule
What is the sine rule, and when do you use it?
Answer it in your head, then turn the card over.
The sine rule
Key points
- Use it in any triangle when you know a side and the angle opposite it, plus one more.
- Each side is paired with the angle opposite it: side a with angle A.
- Use the upside-down form to find an angle.
Formulae
a/sin A = b/sin B = c/sin C
sin A/a = sin = sin C/c
a = 7 cm, A = 40°, B = 65°; find b
b/sin 65° = 7/sin 40°
b = 7 × sin 65° ÷ sin 40° = 9.87 cm (2 d.p.)
Watch out
Pair each side with the angle opposite it, never an angle next to it.
Trigonometry
HigherThe cosine rule for a side
How do you use the cosine rule to find a missing side?
Answer it in your head, then turn the card over.
The cosine rule for a side
Key points
- Use it when you know two sides and the angle between them.
- Angle A sits between sides b and c; side a is opposite it.
- Work out 2bc cos A as one product, then subtract it.
Formula
a² = b² + c² − 2bc cos A
b = 6 cm, c = 9 cm, A = 50°
a² = 36 + 81 − 2 × 6 × 9 × cos 50°
= 117 − 69.42 = 47.58
a = = 6.90 cm (2 d.p.)
Watch out
Never work out (b² + c² − 2bc) first and then multiply by cos A: multiply, then subtract.
Trigonometry
HigherThe cosine rule for an angle
How do you find an angle when you know all three sides of a triangle?
Answer it in your head, then turn the card over.
The cosine rule for an angle
Key points
- Use the rearranged cosine rule.
- a is the side opposite the angle you want; b and c are the other two.
- Work out the fraction, then use cos⁻¹.
Formula
cos A = (b² + c² − a²) ÷ 2bc
Sides 5, 7 and 8; angle opposite the 7
cos A = (5² + 8² − 7²) ÷ (2 × 5 × 8) = =
A = cos⁻¹ = 60°
Watch out
The side subtracted is the one opposite the angle you want, here the 7.
Trigonometry
HigherArea = ½ab sin C
How do you find the area of a triangle from two sides and the angle between them?
Answer it in your head, then turn the card over.
Area = ½ab sin C
Key points
- Area = ½ab sin C, where C is the angle between sides a and b.
- It is ½ × base × height in disguise: b sin C is the height on base a.
- It works in any triangle, with or without a right angle.
Formula
Area = ½ab sin C
Sides 7 cm and 10 cm, 30° between them
½ × 7 × 10 = 35
Area = 35 × sin 30° = 35 × 0.5 = 17.5 cm²
Watch out
The angle must be the one between the two sides, and the ½ is easy to leave out.
Trigonometry
HigherWhich rule to use
In a triangle without a right angle, how do you choose between the sine rule and the cosine rule?
Answer it in your head, then turn the card over.
Which rule to use
Key points
- A side and its opposite angle both known: the sine rule.
- Two sides and the angle between them, to find the third side: the cosine rule.
- All three sides, to find an angle: the cosine rule.
- Area from two sides and the angle between them: ½ab sin C.
- A right angle: SOH CAH TOA or Pythagoras is quicker.
Watch out
Look for a side and its opposite angle first: no such pair means the cosine rule.
Trigonometry
HigherTrigonometry in 3D
How do you find lengths and angles inside a 3D shape such as a cuboid?
Answer it in your head, then turn the card over.
Trigonometry in 3D
Key points
- Find a right-angled triangle inside the solid and sketch it flat, with its own labels.
- The space diagonal: Pythagoras across the base first, then again with the height.
- The angle with the base sits in the triangle of the space diagonal, the base diagonal and the height.
Cuboid 4 by 3 base, height 12
Base diagonal = = 5
Space diagonal = = 13
Angle with the base: tan θ = 12 ÷ 5, so θ = 67.4°
Watch out
The angle with the base is next to the base diagonal, not one of the base's edges.
Print these cards
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Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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