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Therefore, AB is opposite angle C."},{"text":"Side AC","correct":false,"explanation":"AC meets angle C at vertex C, so it is adjacent to that angle. The side opposite angle C is AB because it does not touch vertex C."},{"text":"Side BC","correct":false,"explanation":"BC meets angle C at vertex C, making it adjacent rather than opposite. AB is the only side that does not meet vertex C."},{"text":"Side C","correct":false,"explanation":"C labels a vertex and angle here, not one of the triangle’s named sides. The side lying opposite vertex C is AB."}]},{"id":"8407f6ae-8b98-3e9b-347d-c4d1997354f8","cardType":"basic","points":2,"front":"What is the minimum information required to use the sine rule?","back":"

One complete opposite pair (side and angle)
One other side or angle

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

One complete opposite pair (side and angle)
One other side or angle

","correct":true,"explanation":"The sine rule requires one known side and its opposite angle to form a complete ratio, plus another side or angle whose corresponding value can be found."},{"text":"

Two known side lengths
No known angles

","correct":false,"explanation":"Two sides alone do not provide a known sine-rule ratio because neither opposite angle is known. One complete opposite side-angle pair and one additional value are required."},{"text":"

Two known angles
No known sides

","correct":false,"explanation":"Two angles alone determine the triangle’s shape but not its scale, so no side length can be calculated. A complete opposite pair and another value are needed."},{"text":"

One adjacent pair (side and angle)
One other side or angle

","correct":false,"explanation":"An adjacent side-angle pair cannot form a matching sine-rule ratio. The known side and angle must be opposite each other, with one additional side or angle."}]},{"id":"fe968fc2-0552-7e38-8b0b-c80028d9b8ae","cardType":"basic","points":2,"front":"What calculation should be performed first if a triangle has two known angles and one side?","back":"Find the third angle using the $180^\\text{o}$ sum rule.","mcqOptions":[{"text":"Find the third angle using the $180^\\text{o}$ sum rule.","correct":true,"explanation":"The interior angles of a triangle total $180^\\text{o}$. Subtracting the two known angles gives the third angle, which may complete the opposite pair needed for the sine rule."},{"text":"Find the missing side using the sine rule immediately.","correct":false,"explanation":"The sine rule should not be applied before identifying the required opposite side-angle pairs. First find the third angle by subtracting the two known angles from $180^\\text{o}$."},{"text":"Find the angle total by adding the two known angles.","correct":false,"explanation":"Adding the known angles does not directly give the missing angle. The third angle equals $180^\\text{o}$ minus the sum of the two known angles."},{"text":"Find the third angle using the $90^\\text{o}$ sum rule.","correct":false,"explanation":"Using $90^\\text{o}$ incorrectly assumes that the triangle is right-angled. For any triangle, subtract the two known angles from the full $180^\\text{o}$ angle sum."}]},{"id":"3bad592d-8dd9-90d7-5337-3cc7ce80e806","cardType":"cloze","points":null,"front":"When using the sine rule, ensure your calculator is in [ ] mode.","back":"When using the sine rule, ensure your calculator is in degrees mode.","mcqOptions":[{"text":"degrees","correct":true,"explanation":"GCSE triangle angles are measured in degrees unless stated otherwise, so degree mode ensures that sine values and inverse-sine results use the correct angular unit."},{"text":"radians","correct":false,"explanation":"Radians are a different unit of angular measure and would produce unsuitable sine values for angles given in degrees. The calculator must be set to degrees mode."},{"text":"gradians","correct":false,"explanation":"Gradians divide a full turn into 400 units rather than 360, so they do not match GCSE angles stated in degrees. Use degrees mode instead."},{"text":"standard form","correct":false,"explanation":"Standard form controls how numbers are displayed, not how trigonometric angles are interpreted. The calculator’s angular setting must be degrees mode."}]},{"id":"cc57f513-f194-7324-cb32-d3084cb03cbc","cardType":"basic","points":1,"front":"When calculating a missing side, how should the sine rule formula be arranged?","back":"Side lengths on top: $\\frac{a}{\\sin A} = \\frac{b}{\\sin B}$","mcqOptions":[{"text":"Side lengths on top: $\\frac{a}{\\sin A} = \\frac{b}{\\sin B}$","correct":true,"explanation":"Placing each side above the sine of its opposite angle makes it straightforward to rearrange for an unknown side while preserving the correct opposite pairings."},{"text":"Side lengths on top: $\\frac{a}{\\sin B} = \\frac{b}{\\sin A}$","correct":false,"explanation":"This crosses the opposite pairs: side $a$ belongs with angle $A$, while side $b$ belongs with angle $B$. The correct ratios are $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}$."},{"text":"Angles on top: $\\frac{A}{a} = \\frac{B}{b}$","correct":false,"explanation":"The sine rule uses the sine of each angle, not the raw angle itself. For a missing side, use $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}$."},{"text":"Side lengths on top: $\\frac{a}{\\cos A} = \\frac{b}{\\cos B}$","correct":false,"explanation":"Using cosine confuses the sine rule with cosine-based methods. Each side must be divided by the sine of its opposite angle: $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}$."}]},{"id":"c3e977c7-3f4e-5000-6dbd-edf02ae1d0b6","cardType":"basic","points":1,"front":"When calculating a missing angle, how should the sine rule formula be arranged?","back":"Sines on top: $\\frac{\\sin A}{a} = \\frac{\\sin B}{b}$","mcqOptions":[{"text":"Sines on top: $\\frac{\\sin A}{a} = \\frac{\\sin B}{b}$","correct":true,"explanation":"Placing each angle’s sine above its opposite side allows the equation to be rearranged to isolate the sine of the missing angle before using inverse sine."},{"text":"Sines on top: $\\frac{\\sin A}{b} = \\frac{\\sin B}{a}$","correct":false,"explanation":"The denominators have been paired with the wrong angles. Angle $A$ is opposite side $a$ and angle $B$ is opposite side $b$, giving $\\frac{\\sin A}{a}=\\frac{\\sin B}{b}$."},{"text":"Cosines on top: $\\frac{\\cos A}{a} = \\frac{\\cos B}{b}$","correct":false,"explanation":"The sine rule requires sine rather than cosine, so this applies the wrong trigonometric function. The correct arrangement is $\\frac{\\sin A}{a}=\\frac{\\sin B}{b}$."},{"text":"Angles on top: $\\frac{A}{a} = \\frac{B}{b}$","correct":false,"explanation":"Using raw angle values instead of their sines is not the sine rule. The required ratios are $\\frac{\\sin A}{a}=\\frac{\\sin B}{b}$."}]},{"id":"0bef8ee9-93fc-07e8-c634-c6bcddabfe0f","cardType":"cloze","points":null,"front":"To find a missing side, the formula is: $\\frac{\\text{[...]}}{\\sin A} = \\frac{b}{\\sin B}$","back":"To find a missing side, the formula is: $\\frac{\\boxed{a}}{\\sin A} = \\frac{b}{\\sin B}$","mcqOptions":[{"text":"$$a$","correct":true,"explanation":"The sine rule pairs each side with the sine of its opposite angle. Side $a$ is opposite angle $A$, so the numerator above $\\sin A$ must be $a$."},{"text":"$$b$","correct":false,"explanation":"Using $b$ above $\\sin A$ incorrectly pairs side $b$ with angle $A$. Side $b$ is opposite $B$, while the side paired with $\\sin A$ is $a$."},{"text":"$$c$","correct":false,"explanation":"Using $c$ confuses the three corresponding side-angle pairs. Side $c$ belongs with $\\sin C$; because the denominator is $\\sin A$, the numerator must be $a$."},{"text":"$$d$","correct":false,"explanation":"A triangle conventionally labels its three opposite side-angle pairs as $a,A$, $b,B$, and $c,C$. There is no corresponding $d$ here, so $\\sin A$ must pair with $a$."}]},{"id":"a9d3f0e5-e055-5a59-29c5-2f5377bae91a","cardType":"cloze","points":null,"front":"To find a missing side, the formula is: $\\frac{a}{\\text{[...]}} = \\frac{b}{\\sin B}$","back":"To find a missing side, the formula is: $\\frac{a}{\\boxed{\\sin A}} = \\frac{b}{\\sin B}$","mcqOptions":[{"text":"$$\\sin A$","correct":true,"explanation":"Side $a$ is opposite angle $A$, so its denominator must be the sine of that angle. This creates the corresponding ratio $\\frac{a}{\\sin A}$."},{"text":"$$A$","correct":false,"explanation":"The sine rule uses the sine of an angle, not the angle value itself. The denominator paired with side $a$ must therefore be $\\sin A$."},{"text":"$$\\cos A$","correct":false,"explanation":"Using cosine confuses the sine rule with a different trigonometric relationship. Side $a$ must be divided by the sine of its opposite angle, $\\sin A$."},{"text":"$$\\sin B$","correct":false,"explanation":"Angle $B$ is opposite side $b$, not side $a$. Keeping opposite pairs together means the denominator beneath $a$ must be $\\sin A$."}]},{"id":"f1f40da1-8cf4-6103-fb16-d42d4487670a","cardType":"cloze","points":null,"front":"To find a missing angle, the formula is: $\\frac{\\text{[...]}}{a} = \\frac{\\sin B}{b}$","back":"To find a missing angle, the formula is: $\\frac{\\boxed{\\sin A}}{a} = \\frac{\\sin B}{b}$","mcqOptions":[{"text":"$$\\sin A$","correct":true,"explanation":"The sine rule pairs side $a$ with its opposite angle $A$, so the corresponding numerator is $\\sin A$."},{"text":"$$\\cos A$","correct":false,"explanation":"This confuses the sine rule with a cosine relationship. The numerator must be $\\sin A$ because each side is paired with the sine of its opposite angle."},{"text":"$$\\sin B$","correct":false,"explanation":"This incorrectly pairs side $a$ with angle $B$. The sine rule matches each side with its opposite angle, so the numerator must be $\\sin A$."},{"text":"$$\\cos B$","correct":false,"explanation":"This uses both the wrong trigonometric function and the wrong angle. Side $a$ must be paired with $\\sin A$ in the sine rule."}]},{"id":"c37c0ed2-6bf7-ee5d-5631-8ec989dc22ec","cardType":"cloze","points":null,"front":"To find a missing angle, the formula is: $\\frac{\\sin A}{\\text{[...]}} = \\frac{\\sin B}{b}$","back":"To find a missing angle, the formula is: $\\frac{\\sin A}{\\boxed{a}} = \\frac{\\sin B}{b}$","mcqOptions":[{"text":"$$a$","correct":true,"explanation":"Angle $A$ is opposite side $a$, so the sine rule forms the corresponding ratio $\\frac{\\sin A}{a}$."},{"text":"$$b$","correct":false,"explanation":"This pairs angle $A$ with the side opposite angle $B$. Each sine must be paired with its corresponding opposite side, so the denominator is $a$."},{"text":"$$2a$","correct":false,"explanation":"This introduces an unnecessary factor of two, perhaps confused with the triangle area formula. The side opposite angle $A$ is simply $a$."},{"text":"$$c$","correct":false,"explanation":"This incorrectly pairs angle $A$ with the side opposite angle $C$. In standard triangle notation, angle $A$ is opposite side $a$."}]},{"id":"5eb08dc2-f778-f017-a296-7b2a11333553","cardType":"basic","points":1,"front":"What is the purpose of the inverse sine ($\\sin^{-1}$) function in this topic?","back":"To find an angle from its sine value.","mcqOptions":[{"text":"To find an angle from its sine value.","correct":true,"explanation":"After rearranging the sine rule to obtain $\\sin A$, applying $\\sin^{-1}$ gives the corresponding angle $A$."},{"text":"To find a side length from an angle.","correct":false,"explanation":"Finding a side length requires rearranging the sine rule, not applying inverse sine alone. The inverse sine function converts a known sine value into an angle."},{"text":"To find a sine value from an angle.","correct":false,"explanation":"This describes the ordinary sine function rather than inverse sine. The inverse function $\\sin^{-1}$ takes a sine value and returns the corresponding angle."},{"text":"To find a cosine value from an angle.","correct":false,"explanation":"Cosine values are calculated using the cosine function, not inverse sine. In this topic, $\\sin^{-1}$ is used to recover an angle from its sine value."}]},{"id":"e4dc88b7-d2da-d049-dc27-82edda330e0b","cardType":"basic","points":2,"front":"Why should side lengths be kept as full calculator values until the final step?","back":"To prevent rounding errors in the final perimeter or area.","mcqOptions":[{"text":"To prevent rounding errors in the final perimeter or area.","correct":true,"explanation":"Rounding intermediate side lengths can carry inaccuracies into later addition or area calculations. Keeping full calculator values preserves accuracy until the final answer is rounded."},{"text":"To prevent unit errors in the final perimeter or area.","correct":false,"explanation":"Units must be handled consistently, but retaining full calculator values does not prevent unit errors. It prevents intermediate rounding from reducing the accuracy of the final result."},{"text":"To prevent angle errors in the final perimeter or area.","correct":false,"explanation":"Angle errors arise from using the wrong angle or calculator mode, not from storing full side values. Full values are retained to minimise accumulated rounding errors."},{"text":"To prevent formula errors in the final perimeter or area.","correct":false,"explanation":"Keeping extra decimal places cannot correct a wrongly selected formula. Its purpose is to avoid accumulating rounding errors before the final perimeter or area is calculated."}]},{"id":"bc13b4f7-e186-d612-0378-8b15ee3b71af","cardType":"basic","points":2,"front":"What are the two steps to find the perimeter of a triangle using the sine rule?","back":"

Calculate all missing side lengths
Add all three sides together

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

Calculate all missing side lengths
Add all three sides together

","correct":true,"explanation":"Perimeter is the total distance around a shape. After the sine rule supplies any unknown side lengths, all three sides of the triangle must be added."},{"text":"

Calculate all missing angle sizes
Add all three angles together

","correct":false,"explanation":"Adding the angles confirms their total is $180^\\circ$, but it does not give perimeter. The missing side lengths must be calculated and all three sides added."},{"text":"

Calculate all missing side lengths
Multiply all three sides together

","correct":false,"explanation":"Multiplying the side lengths does not measure the boundary of a triangle. Once missing sides are found, perimeter is calculated by adding all three side lengths."},{"text":"

Calculate all missing side lengths
Add only two sides together

","correct":false,"explanation":"This omits one edge of the triangle, so it cannot give the complete distance around it. Perimeter requires the sum of all three side lengths."}]},{"id":"072e97f9-ebc1-0e96-2360-c4746c3d2cb8","cardType":"basic","points":1,"front":"What is the formula for the area of a non-right-angled triangle?","back":"$$\\text{Area} = \\frac{1}{2}ab\\sin C$","mcqOptions":[{"text":"$$\\text{Area} = \\frac{1}{2}ab\\sin C$","correct":true,"explanation":"Two side lengths and their included angle determine the area. Multiplying $ab$ by $\\sin C$ gives the corresponding perpendicular component, then halving gives the triangle’s area."},{"text":"$$\\text{Area} = ab\\sin C$","correct":false,"explanation":"This misses the factor of $\\frac{1}{2}$, doubling the triangle’s area. The correct formula is $\\text{Area} = \\frac{1}{2}ab\\sin C$."},{"text":"$$\\text{Area} = \\frac{1}{2}ab\\cos C$","correct":false,"explanation":"This substitutes cosine for sine, confusing the area formula with relationships from the cosine rule. The area uses $\\sin C$ with the two enclosing sides."},{"text":"$$\\text{Area} = \\frac{1}{2}(a + b)\\sin C$","correct":false,"explanation":"The side lengths must be multiplied, not added, because area is based on base times perpendicular height. The correct formula is $\\frac{1}{2}ab\\sin C$."}]},{"id":"e926eab3-c0db-7fa3-886d-4ad6918df3d3","cardType":"basic","points":1,"front":"In the area formula $\\frac{1}{2}ab\\sin C$, what does C represent?","back":"The included angle (the angle between sides $a$ and $b$).","mcqOptions":[{"text":"The included angle (the angle between sides $a$ and $b$).","correct":true,"explanation":"In $\\frac{1}{2}ab\\sin C$, angle $C$ must lie between the two side lengths being multiplied, $a$ and $b$."},{"text":"The included angle (the angle between sides $a$ and $c$).","correct":false,"explanation":"The angle between sides $a$ and $c$ is angle $B$ in standard notation. The formula shown uses angle $C$, which lies between sides $a$ and $b$."},{"text":"The included angle (the angle between sides $b$ and $c$).","correct":false,"explanation":"The angle between sides $b$ and $c$ is angle $A$, not angle $C$. Here, $C$ is the included angle between sides $a$ and $b$."},{"text":"The exterior angle (the angle outside sides $a$ and $b$).","correct":false,"explanation":"The standard area formula uses the interior included angle between the two known sides. Therefore, $C$ is the interior angle between $a$ and $b$."}]},{"id":"92aa19f7-4edb-a52f-7bfb-c0d45f111255","cardType":"cloze","points":null,"front":"In the ambiguous case, the identity used to find obtuse angles is $\\sin \\theta = \\sin(\\text{[...]})$.","back":"In the ambiguous case, the identity used to find obtuse angles is $\\sin \\theta = \\sin(\\boxed{180^\\circ - \\theta})$.","mcqOptions":[{"text":"$$180^\\circ - \\theta$","correct":true,"explanation":"Sine is positive in the first and second quadrants, so an acute angle $\\theta$ and its supplementary obtuse angle $180^\\circ - \\theta$ have equal sine values."},{"text":"$$360^\\circ - \\theta$","correct":false,"explanation":"This gives a reflex angle with the opposite sine sign for angles between $0^\\circ$ and $180^\\circ$. The supplementary angle is found using $180^\\circ - \\theta$."},{"text":"$$90^\\circ - \\theta$","correct":false,"explanation":"This is the complementary angle and its sine generally equals $\\cos\\theta$, not $\\sin\\theta$. The second possible triangle angle is $180^\\circ - \\theta$."},{"text":"$$180^\\circ + \\theta$","correct":false,"explanation":"Adding $180^\\circ$ changes the sign of the sine, so it does not produce the same positive sine value. The obtuse alternative is $180^\\circ - \\theta$."}]},{"id":"72c19980-ff4e-46da-10f7-19c6db84aabc","cardType":"basic","points":1,"front":"If $\\sin^{-1}$ gives an acute angle of $52.6^\\circ$ but the question states the angle is obtuse, what is the value?","back":"$$180^\\circ - 52.6^\\circ = 127.4^\\circ$","mcqOptions":[{"text":"$$180^\\circ - 52.6^\\circ = 127.4^\\circ$","correct":true,"explanation":"The sine of an acute angle equals the sine of its supplementary obtuse angle. Subtracting $52.6^\\circ$ from $180^\\circ$ therefore gives the required obtuse angle, $127.4^\\circ$."},{"text":"$$180^\\circ + 52.6^\\circ = 232.6^\\circ$","correct":false,"explanation":"Adding the acute angle produces a reflex angle rather than an obtuse one. The supplementary angle is found by subtraction: $180^\\circ - 52.6^\\circ = 127.4^\\circ$."},{"text":"$$90^\\circ + 52.6^\\circ = 142.6^\\circ$","correct":false,"explanation":"Adding the angle to $90^\\circ$ does not produce the supplementary angle with the same sine. Subtract from $180^\\circ$ instead, giving $127.4^\\circ$."},{"text":"$$180^\\circ - 52.6^\\circ = 137.4^\\circ$","correct":false,"explanation":"The correct method is shown, but the subtraction contains an arithmetic error. Calculating it accurately gives $180^\\circ - 52.6^\\circ = 127.4^\\circ$."}]},{"id":"2de7a85e-df7a-4df8-2172-0e0271d1dac1","cardType":"basic","points":1,"front":"How do you identify an opposite side for the sine rule?","back":"It is the side across from the angle, not touching the angle's vertex.","mcqOptions":[{"text":"It is the side across from the angle, not touching the angle's vertex.","correct":true,"explanation":"An opposite side lies directly across the triangle from its corresponding angle. It does not meet that angle's vertex, so the side and angle form a sine-rule pair."},{"text":"It is the side next to the angle, touching the angle's vertex.","correct":false,"explanation":"A side touching the angle's vertex is adjacent, not opposite. The opposite side lies across from the angle and does not touch the angle's vertex."},{"text":"It is the side longest in the triangle, regardless of the angle's vertex.","correct":false,"explanation":"The longest side is opposite the largest angle, but it is not automatically opposite every chosen angle. Identify the side directly across from the specific angle."},{"text":"It is the side shortest in the triangle, regardless of the angle's vertex.","correct":false,"explanation":"The shortest side is opposite the smallest angle, but side length alone does not identify an opposite pair. Use the side directly across from the specified angle."}]},{"id":"ef73f6b6-5e7b-c1f7-adb0-be31d53109d6","cardType":"basic","points":2,"front":"When given side $c$, angle $C$, and side $a$, how do you find angle A?","back":"$$\\sin A = \\frac{a \\sin C}{c}$ followed by $\\sin^{-1}$.","mcqOptions":[{"text":"$$\\sin A = \\frac{a \\sin C}{c}$ followed by $\\sin^{-1}$.","correct":true,"explanation":"From the sine rule, $\\frac{a}{\\sin A}=\\frac{c}{\\sin C}$. Rearranging gives $\\sin A=\\frac{a\\sin C}{c}$, then inverse sine gives the angle."},{"text":"$$\\sin A = \\frac{c \\sin C}{a}$ followed by $\\sin^{-1}$.","correct":false,"explanation":"This reverses the side ratio when rearranging the sine rule. Matching each side with its opposite angle gives $\\sin A=\\frac{a\\sin C}{c}$ before applying inverse sine."},{"text":"$$\\sin A = \\frac{a \\cos C}{c}$ followed by $\\sin^{-1}$.","correct":false,"explanation":"This incorrectly replaces the sine of the known angle with cosine. The sine rule uses sine for both angles, so $\\sin A=\\frac{a\\sin C}{c}$."},{"text":"$$\\sin A = \\frac{c \\cos C}{a}$ followed by $\\sin^{-1}$.","correct":false,"explanation":"This both reverses the required side ratio and uses cosine instead of sine. Correct rearrangement of the sine rule gives $\\sin A=\\frac{a\\sin C}{c}$."}]},{"id":"66a30248-670c-aa5f-cffa-8cfd1cbc09db","cardType":"basic","points":1,"front":"What specific condition triggers the 'ambiguous case' (obtuse angle twist)?","back":"When a calculated angle could be acute or obtuse, and the problem specifies which one to use.","mcqOptions":[{"text":"When a calculated angle could be acute or obtuse, and the problem specifies which one to use.","correct":true,"explanation":"Sine has the same positive value for supplementary acute and obtuse angles. If the question specifies the angle type, this determines whether to use the inverse-sine result or its supplement."},{"text":"When a calculated angle could be acute or right-angled, and the problem specifies which one to use.","correct":false,"explanation":"A positive sine value generally pairs an acute angle with its supplementary obtuse angle, not a right angle. The ambiguity is resolved by choosing acute or obtuse as specified."},{"text":"When a calculated angle could be obtuse or reflex, and the problem specifies which one to use.","correct":false,"explanation":"Interior angles of a triangle cannot be reflex, so this is not the sine-rule ambiguity. The possible supplementary solutions within the relevant range are acute and obtuse."},{"text":"When a calculated angle could be equal or complementary, and the problem specifies which one to use.","correct":false,"explanation":"Complementary angles add to $90^\\circ$ and do not generally have equal sines. The ambiguous case concerns supplementary acute and obtuse angles, which add to $180^\\circ$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"The Sine Rule","title":"The Sine Rule Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"595e2658-f1a3-48a4-a7a3-759eb5433a36","walled":"$L37"}],"$L38","$L39"]