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set up the proportion."},{"text":"A complete adjacent pair (a side and an angle beside it).","correct":false,"explanation":"The sine rule does not use a side and an adjacent angle; it requires an opposite pair. The side must be matched with the angle directly across from it."},{"text":"Two equal sides and the angle between them.","correct":false,"explanation":"Two equal sides and the included angle describe an isosceles-triangle situation, but they are not required for the sine rule. One known opposite side-angle pair is essential."},{"text":"A complete right-angle pair (the hypotenuse and a right angle).","correct":false,"explanation":"The sine rule applies to any triangle and does not require a right angle. What is essential is a side paired with the angle directly opposite it."}]},{"id":"a7eb4348-ceaf-4e04-a31b-1ca439687741","cardType":"basic","points":1,"front":"How do you identify an opposite side for an angle in a triangle?","back":"It is the side directly across from the angle (it does not touch the angle).","mcqOptions":[{"text":"It is the side directly across from the angle (it does not touch the angle).","correct":true,"explanation":"An opposite side lies across the triangle from the angle and shares neither endpoint with it. This is the side that must be paired with that angle in the sine rule."},{"text":"It is the side directly beside the angle (it touches the angle).","correct":false,"explanation":"A side touching the angle is adjacent, not opposite. The opposite side is directly across the triangle and does not touch the angle."},{"text":"It is the side directly across from the angle (it touches the angle).","correct":false,"explanation":"The side across from an angle cannot touch that angle; touching it would make the side adjacent. The correct description says it does not touch the angle."},{"text":"It is the side longest in the triangle (it does not touch the angle).","correct":false,"explanation":"The longest side is only opposite the largest angle, so it is not automatically opposite every chosen angle. Opposite means directly across from the specific angle."}]},{"id":"08a98bf5-5bdc-414b-9f12-fc299ea4e7a6","cardType":"cloze","points":null,"front":"In triangle ABC, the sum of internal angles is $A + B + C = \\text{[...]}$.","back":"In triangle ABC, the sum of internal angles is $A + B + C = \\boxed{180^\\circ}$.","mcqOptions":[{"text":"$$180^\\circ$","correct":true,"explanation":"The three internal angles of every triangle add to $180^\\circ$, so $A + B + C = 180^\\circ$."},{"text":"$$90^\\circ$","correct":false,"explanation":"$$90^\\circ$ is the angle sum associated with a straight angle split incorrectly here, not a triangle. The internal angles of a triangle always total $180^\\circ$."},{"text":"$$360^\\circ$","correct":false,"explanation":"$$360^\\circ$ is a full turn, not the sum of a triangle’s internal angles. The correct total is $180^\\circ$."},{"text":"$$270^\\circ$","correct":false,"explanation":"$$270^\\circ$ may result from confusing a triangle with three right angles, but triangle angles total $180^\\circ$."}]},{"id":"6707423c-2ecc-47b8-89d7-8003e88ab413","cardType":"basic","points":1,"front":"Write the version of the sine rule formula best suited for finding a missing side.","back":"$$$\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}$$","mcqOptions":[{"text":"$$$\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}$$","correct":true,"explanation":"For finding a missing side, pair each side with the sine of its opposite angle. Equating these ratios lets you solve for the unknown side."},{"text":"$$$\\frac{a}{\\cos A} = \\frac{b}{\\cos B} = \\frac{c}{\\cos C}$$","correct":false,"explanation":"The sine rule uses sines, not cosines. Replacing sine with cosine gives a different and generally invalid relationship, so it will produce the wrong missing side."},{"text":"$$$\\frac{a}{\\sin A} = \\frac{b}{\\sin C} = \\frac{c}{\\sin B}$$","correct":false,"explanation":"The side-angle pairings have been swapped: $b$ must pair with $B$, and $c$ must pair with $C$. Matching each side to its opposite angle gives the correct formula."},{"text":"$$$\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin B}$$","correct":false,"explanation":"Both $b$ and $c$ cannot be paired with $B$ in the general sine rule. Each side needs its own opposite angle, so $c$ must be paired with $C$."}]},{"id":"6da2e531-5b3b-46d7-86cc-0d00d9ad9e81","cardType":"basic","points":1,"front":"Write the version of the sine rule formula best suited for finding a missing angle.","back":"$$$\\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin C}{c}$$","mcqOptions":[{"text":"$$$\\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin C}{c}$$","correct":true,"explanation":"This is the reciprocal form of the sine rule, arranged with sine values over their corresponding opposite sides. It is convenient when solving for a missing angle."},{"text":"$$$\\frac{\\cos A}{a} = \\frac{\\cos B}{b} = \\frac{\\cos C}{c}$$","correct":false,"explanation":"The sine rule requires sine values, not cosine values. Using cosine here changes the relationship and gives an incorrect result for the missing angle."},{"text":"$$$\\frac{\\sin A}{b} = \\frac{\\sin B}{a} = \\frac{\\sin C}{c}$$","correct":false,"explanation":"The first two sine values have been matched with the wrong sides. Each angle must be paired with its opposite side: $A$ with $a$ and $B$ with $b$."},{"text":"$$$\\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin B}{c}$$","correct":false,"explanation":"The final term repeats $B$ instead of using angle $C$. The correct pairing is $C$ with side $c$, because they are opposite each other."}]},{"id":"a14aa0bf-da53-41cb-85f4-dc000c7b8c54","cardType":"basic","points":1,"front":"What calculator mode must be used for sine rule calculations in IGCSE Maths?","back":"Degrees mode.","mcqOptions":[{"text":"Degrees mode.","correct":true,"explanation":"IGCSE triangle angles are normally given in degrees, so the calculator must be in degree mode for sine and inverse-sine calculations to produce the expected answers."},{"text":"Radians mode.","correct":false,"explanation":"Radians are a different angle measure and would interpret a degree input incorrectly. For standard IGCSE sine rule questions, use Degrees mode."},{"text":"Grads mode.","correct":false,"explanation":"Grads are another angle unit and are not the unit normally used in IGCSE triangle questions. Degree mode is required when the angles are given in degrees."},{"text":"Turns mode.","correct":false,"explanation":"Turns measure complete rotations rather than ordinary triangle angles. Entering degree values in turns mode gives incorrect sine values; the correct setting is Degrees mode."}]},{"id":"b24363a0-42b9-4d63-b06f-6f9ee4984df8","cardType":"basic","points":1,"front":"What is the function of the inverse sine ($\\sin^{-1}$) button?","back":"It finds the angle when the sine value is already known.","mcqOptions":[{"text":"It finds the angle when the sine value is already known.","correct":true,"explanation":"The inverse sine button reverses the sine operation: given a sine value, it returns the corresponding angle, provided the calculator is using the correct angle mode."},{"text":"It finds the side when the angle value is already known.","correct":false,"explanation":"Inverse sine does not calculate a side length. It finds an angle from a known sine value; side lengths are found by rearranging the sine rule."},{"text":"It finds the sine value when the angle is already known.","correct":false,"explanation":"The ordinary sine button finds a sine value from an angle. The inverse sine button works in the opposite direction, finding the angle from the sine value."},{"text":"It finds the angle when the cosine value is already known.","correct":false,"explanation":"A cosine value must be processed with inverse cosine, not inverse sine. Inverse sine specifically finds an angle from a known sine value."}]},{"id":"4dd82ea6-a8d9-46fc-b18f-e243f23d3a85","cardType":"basic","points":2,"front":"Why should you avoid rounding the sine value halfway through a calculation?","back":"To prevent rounding errors in the final answer; keep the full value until the end.","mcqOptions":[{"text":"To prevent rounding errors in the final answer; keep the full value until the end.","correct":true,"explanation":"Rounding an intermediate sine value changes the number used in later calculations. Keeping full calculator precision reduces accumulated error and gives a more accurate final answer."},{"text":"To prevent angle errors in the calculator mode; keep the full value until the end.","correct":false,"explanation":"Rounding does not change the calculator’s angle mode. It changes the numerical sine value used later, causing rounding errors in the final answer; keep full precision instead."},{"text":"To prevent unit errors in the side length; keep the full value until the end.","correct":false,"explanation":"Units are controlled by the measurements and formula, not by rounding the sine value. Early rounding can cause numerical rounding errors, so retain the full value."},{"text":"To prevent inverse errors in the angle; keep the full value until the end.","correct":false,"explanation":"The inverse sine button is not made more or less valid by delaying rounding. The real issue is that rounding an intermediate value introduces error into the final calculation."}]},{"id":"e6521643-b4c3-4fbc-a71d-80cb49ce0d42","cardType":"basic","points":2,"front":"If you know two angles and one side, but the side is not opposite either angle, what is the first step?","back":"Calculate the third angle using $180^\\circ - (\\text{sum of known angles})$.","mcqOptions":[{"text":"Calculate the third angle using $180^\\circ - (\\text{sum of known angles})$.","correct":true,"explanation":"The third angle creates an opposite angle-side pair with the known side, allowing the sine rule to be applied correctly. Triangle angles always sum to $180^2$39."},{"text":"Calculate the third angle using $360^2 - (\\text{sum of known angles})$.","correct":false,"explanation":"Using $360^2$ confuses the angle sum of a triangle with the angle sum around a point. The third angle is found using $180^2 - (\\text{sum of known angles})$."},{"text":"Calculate the missing side using Pythagoras’ theorem.","correct":false,"explanation":"Pythagoras applies only to right-angled triangles and cannot generally find this missing side. First calculate the third angle so the known side has a known opposite angle."},{"text":"Calculate the missing side using the cosine rule.","correct":false,"explanation":"The cosine rule is unnecessary when two angles and one side are known. First calculate the third angle, creating the opposite angle-side pair needed for the sine rule."}]},{"id":"5c80f035-530b-48e0-9fc0-87f80bab1f9b","cardType":"cloze","points":null,"front":"The formula for the area of a non-right-angled triangle is $\\text{Area} = \\text{[...]}$.","back":"The formula for the area of a non-right-angled triangle is $\\text{Area} = \\boxed{\\frac{1}{2}ab\\sin C}$.","mcqOptions":[{"text":"$$\\frac{1}{2}ab\\sin C$","correct":true,"explanation":"Two sides and their included angle determine the area: multiplying $a$ and $b$ by $\\sin C$ and halving gives the perpendicular-height contribution."},{"text":"$$\\frac{1}{2}ab\\cos C$","correct":false,"explanation":"Using cosine confuses the area formula with relationships appearing in the cosine rule. The triangle’s area is $\\frac{1}{2}ab\\sin C$, where $C$ is the included angle."},{"text":"$$ab\\sin C$","correct":false,"explanation":"Omitting the factor of $\\frac{1}{2}$ doubles the calculated area. The correct formula is $\\frac{1}{2}ab\\sin C$ because a triangle is half the corresponding parallelogram."},{"text":"$$\\frac{ab}{2\\sin C}$","correct":false,"explanation":"Dividing by $\\sin C$ reverses its role in the perpendicular height. The correct area formula multiplies by the sine: $\\frac{1}{2}ab\\sin C$."}]},{"id":"147fa16d-933e-4abe-b1a6-09c6a73b2170","cardType":"basic","points":1,"front":"In the area formula $\\frac{1}{2}ab\\sin C$, what is the relationship between the angle $C$ and sides $a$ and $b$?","back":"$$C$ must be the included angle (the angle between sides $a$ and $b$).","mcqOptions":[{"text":"$$C$ must be the included angle (the angle between sides $a$ and $b$).","correct":true,"explanation":"The height relative to one side is obtained from the other side using $\\sin C$, so $C$ must be the angle directly between sides $a$ and $b$."},{"text":"$$C$ must be an opposite angle (the angle opposite either side $a$ or $b$).","correct":false,"explanation":"Choosing an angle opposite either given side confuses the area formula with sine-rule pairings. In $\\frac{1}{2}ab\\sin C$, $C$ must lie between sides $a$ and $b$."},{"text":"$$C$ must be the exterior angle (the angle outside sides $a$ and $b$).","correct":false,"explanation":"The area formula uses an interior included angle, not an exterior angle. Specifically, $C$ must be the interior angle between sides $a$ and $b$."},{"text":"$$C$ must be the right angle (a $90^2$ angle between sides $a$ and $b$).","correct":false,"explanation":"The triangle need not be right-angled, so assuming $C=90^2$ is an unjustified restriction. $C$ is simply the included angle between sides $a$ and $b$."}]},{"id":"91c5252e-3442-46aa-bd7a-e42e543f84fc","cardType":"basic","points":1,"front":"What is the trigonometric identity linking the sine of an acute angle and its obtuse supplement?","back":"$$$\\sin \\theta = \\sin(180^\\circ - \\theta)$$","mcqOptions":[{"text":"$$$\\sin \\theta = \\sin(180^\\circ - \\theta)$$","correct":true,"explanation":"Supplementary angles have equal sine values because they have the same vertical component on the unit circle. Therefore an acute angle and its obtuse supplement share the same sine."},{"text":"$$$\\sin \\theta = -\\sin(180^2 - \\theta)$$","correct":false,"explanation":"This sign change is a common unit-circle error. Sine is positive throughout the first and second quadrants, so $\\sin(180^2-\\theta)=\\sin\\theta$, not its negative."},{"text":"$$$\\sin \\theta = \\cos(180^2 - \\theta)$$","correct":false,"explanation":"This incorrectly substitutes cosine for sine. Supplementary angles have equal sine values, giving $\\sin\\theta=\\sin(180^2-\\theta)$; their cosine values instead have opposite signs."},{"text":"$$$\\sin \\theta = \\sin(90^2 - \\theta)$$","correct":false,"explanation":"This confuses supplementary-angle and complementary-angle identities. The complementary identity is $\\sin\\theta=\\cos(90^2-\\theta)$, while the obtuse supplement gives $\\sin\\theta=\\sin(180^2-\\theta)$."}]},{"id":"93dcec14-d034-4c79-afe8-62693609f55c","cardType":"basic","points":1,"front":"Define an obtuse angle.","back":"An angle greater than $90^\\circ$ and less than $180^\\circ$.","mcqOptions":[{"text":"An angle greater than $90^\\circ$ and less than $180^\\circ$.","correct":true,"explanation":"An obtuse angle lies strictly between a right angle and a straight angle. Therefore it is greater than $90^2$ but less than $180^2$."},{"text":"An angle greater than $0^2$ and less than $90^2$.","correct":false,"explanation":"Angles strictly between $0^2$ and $90^2$ are acute, not obtuse. An obtuse angle must instead be greater than $90^2$ and less than $180^2$."},{"text":"An angle greater than $180^2$ and less than $360^2$.","correct":false,"explanation":"Angles strictly between $180^2$ and $360^2$ are reflex angles. An obtuse angle lies between $90^2$ and $180^2$ instead."},{"text":"An angle greater than $90^2$ and at most $180^2$.","correct":false,"explanation":"Including $180^2$ is a boundary error because $180^2$ is a straight angle, not an obtuse angle. The upper boundary must be strictly less than $180^2$."}]},{"id":"061cab71-2c9c-46c7-afe4-456712f1e867","cardType":"basic","points":1,"front":"How do you calculate the obtuse version of an angle if your calculator gives an acute result?","back":"Subtract the acute angle from $180^\\circ$.","mcqOptions":[{"text":"Subtract the acute angle from $180^\\circ$.","correct":true,"explanation":"An acute angle and its obtuse supplement have the same sine. Since supplementary angles sum to $180^2$, subtracting the acute result gives the required obtuse angle."},{"text":"Subtract the acute angle from $90^2$.","correct":false,"explanation":"Subtracting from $90^2$ finds a complementary angle, not the obtuse solution with the same sine. The required supplement is found by subtracting from $180^2$."},{"text":"Add the acute angle to $180^2$.","correct":false,"explanation":"Adding the acute angle to $180^2$ produces a reflex angle greater than $180^2$. The corresponding obtuse solution is $180^2$ minus the acute angle."},{"text":"Subtract the acute angle from $360^2$.","correct":false,"explanation":"Subtracting from $360^2$ gives a reflex angle, reflecting full-turn reasoning rather than supplementary angles. The obtuse solution is found by subtracting from $180^2$."}]},{"id":"67179288-6538-44e8-b281-5317cddcce86","cardType":"basic","points":1,"front":"How do you calculate the perimeter of a triangle after using the sine rule to find missing sides?","back":"Add all three side lengths together.","mcqOptions":[{"text":"Add all three side lengths together.","correct":true,"explanation":"The perimeter is the total distance around the triangle, so it is found by adding the lengths of all three sides after any missing lengths are calculated."},{"text":"Multiply all three side lengths together.","correct":false,"explanation":"Multiplying side lengths does not measure the boundary and produces inappropriate squared or cubed units. Perimeter is found by adding all three side lengths."},{"text":"Add the two longest side lengths together.","correct":false,"explanation":"Adding only two sides omits one part of the triangle’s boundary. The perimeter requires the sum of all three side lengths, including the newly calculated side."},{"text":"Average all three side lengths together.","correct":false,"explanation":"Dividing the total by three calculates the mean side length rather than the perimeter. The perimeter is simply the sum of all three side lengths."}]},{"id":"af053c64-6f61-4f80-9970-d7a46a376dab","cardType":"cloze","points":null,"front":"When finding a side length using the sine rule, it is easiest to put the [ ] on top of the fractions.","back":"When finding a side length using the sine rule, it is easiest to put the side lengths on top of the fractions.","mcqOptions":[{"text":"side lengths","correct":true,"explanation":"Placing side lengths on top gives $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}$, making an unknown side the numerator and easier to isolate."},{"text":"angles","correct":false,"explanation":"Putting angles themselves on top is invalid because the sine rule uses their sine values, not the angles directly. For finding a side, place the side lengths on top."},{"text":"sines of the angles","correct":false,"explanation":"Putting the sines on top is a valid rearrangement, but it is less direct for finding a side because the unknown side appears in a denominator. Side lengths should go on top."},{"text":"opposite angles","correct":false,"explanation":"The sine rule pairs each side with its opposite angle, but opposite angles are not placed as raw quantities in the numerators. Put the side lengths on top."}]},{"id":"fe4e322d-7fa4-4045-8918-f06be06da22c","cardType":"cloze","points":null,"front":"When finding an angle using the sine rule, it is easiest to put the [ ] on top of the fractions.","back":"When finding an angle using the sine rule, it is easiest to put the sine parts on top of the fractions.","mcqOptions":[{"text":"sine parts","correct":true,"explanation":"Placing the sine parts in the numerators makes the unknown angle’s sine easier to isolate before applying the inverse sine function."},{"text":"side lengths","correct":false,"explanation":"Side lengths are usually placed on top when finding an unknown side. For an unknown angle, the sine parts should be on top so the sine can be isolated easily."},{"text":"angle values","correct":false,"explanation":"The angle itself does not appear directly in the sine rule; its sine does. The correct fill is sine parts, which are placed on top when finding an angle."},{"text":"cosine parts","correct":false,"explanation":"The sine rule relates sides to their opposite sines, not cosines. The correct fill is sine parts because these should be on top when finding an angle."}]},{"id":"537d9a48-d303-4ce5-9c0e-09c367efb411","cardType":"basic","points":3,"front":"What information is required to find the area of a triangle if you only know one side and two angles?","back":"

Use the sine rule to find a second side length.
Use the area formula with the two sides and their included angle.

","mcqOptions":[{"text":"

Use the sine rule to find a second side length.
Use the area formula with the two sides and their included angle.

","correct":true,"explanation":"The sine rule uses the known side-angle pair to calculate another side. The triangle’s area can then be found using two sides and the sine of their included angle."},{"text":"

Use the cosine rule to find a second side length.
Use the area formula with the two sides and their included angle.

","correct":false,"explanation":"The cosine rule cannot determine a second side from only one side and two angles. Use the sine rule first, then the area formula with two sides and their included angle."},{"text":"

Use the sine rule to find a second side length.
Use the area formula with the two sides and an opposite angle.

","correct":false,"explanation":"The formula $A=\\frac{1}{2}ab\\sin C$ requires the angle between the two chosen sides, not an opposite angle. First find another side using the sine rule."},{"text":"

Use Pythagoras’ theorem to find a second side length.
Use the area formula with the two sides and their included angle.

","correct":false,"explanation":"Pythagoras applies only to right-angled triangles and cannot generally find the second side here. Use the sine rule, then use the area formula with the included angle."}]},{"id":"e7bde394-3333-4007-bac3-4892d1095bdd","cardType":"basic","points":2,"front":"If side $x$ is opposite $40^\\circ$ and side $10$ is opposite $80^\\circ$, set up the sine rule to find $x$.","back":"$$$\\frac{x}{\\sin 40^\\circ} = \\frac{10}{\\sin 80^\\circ}$$","mcqOptions":[{"text":"$$$\\frac{x}{\\sin 40^\\circ} = \\frac{10}{\\sin 80^\\circ}$$","correct":true,"explanation":"Each side must be paired with the sine of its opposite angle. Thus $x$ pairs with $\\sin 40^\\circ$, while $10$ pairs with $\\sin 80^\\circ$."},{"text":"$$$\\frac{x}{\\sin 80^\\circ} = \\frac{10}{\\sin 40^\\circ}$$","correct":false,"explanation":"This swaps the angles associated with the sides. The correct pairings are $x$ with $40^\\circ$ and $10$ with $80^\\circ$, giving $\\frac{x}{\\sin 40^\\circ}=\\frac{10}{\\sin 80^\\circ}$."},{"text":"$$$\\frac{\\sin 40^\\circ}{x} = \\frac{10}{\\sin 80^\\circ}$$","correct":false,"explanation":"This mixes sine-over-side with side-over-sine, so the ratios are inconsistent. Both fractions must use the same orientation: $\\frac{x}{\\sin 40^\\circ}=\\frac{10}{\\sin 80^\\circ}$."},{"text":"$$$\\frac{x}{10} = \\frac{\\sin 80^\\circ}{\\sin 40^\\circ}$$","correct":false,"explanation":"The sine ratio has been reversed. Dividing the correct sine-rule equation by $10$ gives $\\frac{x}{10}=\\frac{\\sin 40^\\circ}{\\sin 80^\\circ}$, not the opposite ratio."}]},{"id":"ab6a371f-9403-41e6-8350-8a092fd37134","cardType":"basic","points":2,"front":"If $\\sin x = 0.5$ and $x$ is known to be obtuse, what is the value of $x$?","back":"

$\\sin^{-1}(0.5) = 30^\\circ$
$180^\\circ - 30^\\circ = 150^\\circ$

","mcqOptions":[{"text":"

$\\sin^{-1}(0.5) = 30^\\circ$
$180^\\circ - 30^\\circ = 150^\\circ$

","correct":true,"explanation":"The principal inverse sine gives $30^\\circ$. Sine is also positive in the second quadrant, so the required obtuse solution is its supplement, $180^\\circ-30^\\circ=150^\\circ$."},{"text":"

$\\sin^{-1}(0.5) = 30^\\circ$
$180^\\circ + 30^\\circ = 210^\\circ$

","correct":false,"explanation":"Adding the acute angle to $180^\\circ$ produces a reflex angle whose sine is negative. The obtuse solution is the supplement, $180^\\circ-30^\\circ=150^\\circ$."},{"text":"

$\\sin^{-1}(0.5) = 60^\\circ$
$180^\\circ - 60^\\circ = 120^\\circ$

","correct":false,"explanation":"Confusing the inverse sine with the inverse cosine gives the incorrect principal angle of $60^\\circ$. Since $\\sin^{-1}(0.5)=30^\\circ$, the obtuse solution is $150^\\circ$."},{"text":"

$\\sin^{-1}(0.5) = 30^\\circ$
$90^\\circ + 30^\\circ = 120^\\circ$

","correct":false,"explanation":"The second sine solution is found using supplementary angles, not by adding to $90^\\circ$. The correct calculation is $180^\\circ-30^\\circ=150^\\circ$."}]}]}],"examCountdown":null,"heading":"The Sine Rule","title":"The Sine Rule Flashcards","topbar":{"backHref":"/igcse/maths/cie/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"5c28de2b-4160-4e51-9a18-6ff1cac48a40","walled":"$L39"}],"$L3a","$L3b"]