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Fractions and surds, such as $\\frac{1}{2}$ and $\\frac{\\sqrt{3}}{2}$, commonly express exact trigonometric values."},{"text":"An answer written with rounding, typically as a decimal or a percentage.","correct":false,"explanation":"Rounding produces an approximation rather than an exact value. The right answer is written without rounding, usually as a fraction or surd that preserves the complete value."},{"text":"An answer found using a calculator, typically as a decimal to a stated accuracy.","correct":false,"explanation":"A calculator decimal given to a stated accuracy is normally approximate. An exact value is written without rounding, typically using a fraction or a surd."},{"text":"An answer written as a whole number, typically after rounding a fraction or a surd.","correct":false,"explanation":"Rounding a fraction or surd to a whole number loses information, so the result is not exact. Exact values retain the original fraction or surd without rounding."}]},{"id":"d2ad01ec-3ef5-4b09-8f16-39f0f61118de","cardType":"basic","points":1,"front":"Which side of a right-angled triangle is the hypotenuse?","back":"The longest side, which is always opposite the right angle.","mcqOptions":[{"text":"The longest side, which is always opposite the right angle.","correct":true,"explanation":"In every right-angled triangle, the side facing the $90^\\circ$ angle is the hypotenuse. Pythagoras’ theorem also confirms that it must be the longest side."},{"text":"The shortest side, which is always opposite the smallest angle.","correct":false,"explanation":"Although the shortest side is opposite the smallest angle, it is not the hypotenuse. The hypotenuse is the longest side and lies opposite the right angle."},{"text":"The longest side, which is always next to the right angle.","correct":false,"explanation":"This correctly identifies the hypotenuse as longest but places it incorrectly. The two legs form the right angle, while the hypotenuse lies opposite the right angle."},{"text":"The opposite side, which is always opposite the chosen acute angle.","correct":false,"explanation":"The side opposite a chosen acute angle is one of the triangle’s legs, not the hypotenuse. The hypotenuse is specifically opposite the right angle and is longest."}]},{"id":"5bb8d49f-09c4-4b6b-af29-194ce714daf8","cardType":"cloze","points":null,"front":"The side adjacent to an angle $\\theta$ is the side next to it that is not the [ ].","back":"The side adjacent to an angle $\\theta$ is the side next to it that is not the hypotenuse.","mcqOptions":[{"text":"hypotenuse","correct":true,"explanation":"For a chosen acute angle, two sides touch the angle: one leg and the hypotenuse. The adjacent side means the touching leg, so the hypotenuse is excluded."},{"text":"opposite","correct":false,"explanation":"The opposite side lies across from $\\theta$ and does not touch it, so this does not distinguish the adjacent side. The excluded touching side is the hypotenuse."},{"text":"adjacent","correct":false,"explanation":"Excluding the adjacent side would make the definition circular and leave no named leg. The adjacent side is the leg beside $\\theta$ that is not the hypotenuse."},{"text":"shortest side","correct":false,"explanation":"The adjacent leg is not necessarily the shortest side; its length depends on the chosen angle and triangle. It is identified by excluding the hypotenuse."}]},{"id":"e0db7a6b-486a-4117-bce9-9e0f94957407","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the sine ratio as $\\sin\\theta = \\frac{\\text{[...]}}{\\text{hypotenuse}}$.","back":"SOHCAHTOA defines the sine ratio as $\\sin\\theta = \\frac{\\boxed{\\text{opposite}}}{\\text{hypotenuse}}$.","mcqOptions":[{"text":"$$\\text{opposite}$","correct":true,"explanation":"SOH in SOHCAHTOA means sine equals opposite divided by hypotenuse. Therefore, the side opposite $\\theta$ forms the numerator of the sine ratio."},{"text":"$$\\text{adjacent}$","correct":false,"explanation":"Using the adjacent side as the numerator confuses sine with cosine. SOH gives $\\sin\\theta=\\frac{\\text{opposite}}{\\text{hypotenuse}}$, so opposite is the required term."},{"text":"$$\\text{hypotenuse}$","correct":false,"explanation":"Putting the hypotenuse in both numerator and denominator would always give $1$, regardless of the angle. Sine instead uses opposite over hypotenuse."},{"text":"$$\\text{shortest side}$","correct":false,"explanation":"The numerator is determined by its position relative to $\\theta$, not by its length. SOH specifies the opposite side, which is not necessarily the shortest side."}]},{"id":"596b4c15-9c78-409c-8847-5ac22736c644","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the sine ratio as $\\sin\\theta = \\frac{\\text{opposite}}{\\text{[...]}}$.","back":"SOHCAHTOA defines the sine ratio as $\\sin\\theta = \\frac{\\text{opposite}}{\\boxed{\\text{hypotenuse}}}$.","mcqOptions":[{"text":"$$\\text{hypotenuse}$","correct":true,"explanation":"SOH in SOHCAHTOA states that sine is opposite over hypotenuse. The hypotenuse therefore forms the denominator and is always the triangle’s longest side."},{"text":"$$\\text{adjacent}$","correct":false,"explanation":"Opposite divided by adjacent is the tangent ratio, not the sine ratio. Sine follows SOH, so its denominator must be the hypotenuse."},{"text":"$$\\text{opposite}$","correct":false,"explanation":"Using opposite in both positions would make the ratio equal to $1$ for every angle. Sine is opposite divided by the hypotenuse."},{"text":"$$\\text{shortest side}$","correct":false,"explanation":"Sine’s denominator is not selected by comparing side lengths. SOH specifies the hypotenuse, which is always the longest side of a right-angled triangle."}]},{"id":"15b0a271-c34a-4bf0-985b-ae19fb9143c2","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the cosine ratio as $\\cos\\theta = \\frac{\\text{[...]}}{\\text{hypotenuse}}$.","back":"SOHCAHTOA defines the cosine ratio as $\\cos\\theta = \\frac{\\boxed{\\text{adjacent}}}{\\text{hypotenuse}}$.","mcqOptions":[{"text":"$$\\text{adjacent}$","correct":true,"explanation":"CAH in SOHCAHTOA means cosine equals adjacent divided by hypotenuse. The adjacent leg, excluding the hypotenuse, therefore forms the numerator."},{"text":"$$\\text{opposite}$","correct":false,"explanation":"Using opposite over hypotenuse gives sine, not cosine. CAH shows that cosine has the adjacent side as its numerator and the hypotenuse as its denominator."},{"text":"$$\\text{hypotenuse}$","correct":false,"explanation":"Putting the hypotenuse in both positions would make cosine equal to $1$ for every angle. Cosine instead uses adjacent over hypotenuse."},{"text":"$$\\text{shortest side}$","correct":false,"explanation":"The numerator is chosen by its position relative to $\\theta$, not by which leg is shortest. For cosine, CAH specifies the adjacent side."}]},{"id":"301ed0cf-dd1a-4650-9b63-d9691a421e98","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the cosine ratio as $\\cos\\theta = \\frac{\\text{adjacent}}{\\text{[...]}}$.","back":"SOHCAHTOA defines the cosine ratio as $\\cos\\theta = \\frac{\\text{adjacent}}{\\boxed{\\text{hypotenuse}}}$.","mcqOptions":[{"text":"$$\\text{hypotenuse}$","correct":true,"explanation":"CAH in SOHCAHTOA gives $\\cos\\theta=\\frac{\\text{adjacent}}{\\text{hypotenuse}}$. The hypotenuse is therefore the denominator of the cosine ratio."},{"text":"$$\\text{opposite}$","correct":false,"explanation":"Adjacent divided by opposite is the reciprocal of tangent, not cosine. CAH states that cosine’s denominator is the hypotenuse."},{"text":"$$\\text{adjacent}$","correct":false,"explanation":"Using adjacent in both positions would make the ratio equal to $1$ for every angle. Cosine is adjacent divided by the hypotenuse."},{"text":"$$\\text{longest leg}$","correct":false,"explanation":"The denominator is not whichever leg happens to be longer. CAH specifies the hypotenuse, which is not a leg and is always the triangle’s longest side."}]},{"id":"56a57e07-e6d0-4310-8473-ef0e45859d2b","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the tangent ratio as $\\tan\\theta = \\frac{\\text{[...]}}{\\text{adjacent}}$.","back":"SOHCAHTOA defines the tangent ratio as $\\tan\\theta = \\frac{\\boxed{\\text{opposite}}}{\\text{adjacent}}$.","mcqOptions":[{"text":"$$\\text{opposite}$","correct":true,"explanation":"TOA in SOHCAHTOA means tangent equals opposite divided by adjacent. The side opposite $\\theta$ therefore forms the numerator of the tangent ratio."},{"text":"$$\\text{adjacent}$","correct":false,"explanation":"Using adjacent in both positions would make tangent equal to $1$ for every angle. TOA requires the opposite side in the numerator."},{"text":"$$\\text{hypotenuse}$","correct":false,"explanation":"Hypotenuse divided by adjacent is related to the reciprocal of cosine, not tangent. Tangent uses opposite over adjacent and does not involve the hypotenuse."},{"text":"$$\\text{shortest side}$","correct":false,"explanation":"The numerator is determined by its position relative to $\\theta$, not by being the shortest side. TOA identifies it specifically as the opposite side."}]},{"id":"6464541b-b208-402f-af34-cd7dc7ea2e03","cardType":"cloze","points":null,"front":"SOHCAHTOA defines the tangent ratio as $\\tan\\theta = \\frac{\\text{opposite}}{\\text{[...]}}$.","back":"SOHCAHTOA defines the tangent ratio as $\\tan\\theta = \\frac{\\text{opposite}}{\\boxed{\\text{adjacent}}}$.","mcqOptions":[{"text":"$$\\text{adjacent}$","correct":true,"explanation":"Tangent compares the side opposite the angle with the adjacent non-hypotenuse side, so SOHCAHTOA gives $\\tan\\theta=rac{\\text{opposite}}{\\text{adjacent}}$."},{"text":"$$\\text{hypotenuse}$","correct":false,"explanation":"Using the hypotenuse as the denominator gives the sine ratio, not tangent. Tangent is opposite divided by adjacent, so the missing term is $\\text{adjacent}$."},{"text":"$$\\text{opposite}$","correct":false,"explanation":"Placing opposite in both positions would always give a ratio of 1. Tangent compares two different sides: opposite divided by adjacent, so the term is $\\text{adjacent}$."},{"text":"$$\\text{angle}$","correct":false,"explanation":"The angle is used to identify the triangle’s sides, but it is not a side length in the ratio. The denominator of tangent is $\\text{adjacent}$."}]},{"id":"a99e9e63-aea2-44a4-860d-14eb586057ad","cardType":"basic","points":2,"front":"What are the three side lengths of the $45^\\circ$ special triangle?","back":"

Two shorter sides: 1
Hypotenuse: $\\sqrt{2}$

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

Two shorter sides: 1
Hypotenuse: $\\sqrt{2}$

","correct":true,"explanation":"A right-angled isosceles triangle can have legs of length 1. Pythagoras then gives the hypotenuse as $\\sqrt{1^2+1^2}=\\sqrt{2}$."},{"text":"

Two shorter sides: 1
Hypotenuse: 2

","correct":false,"explanation":"A hypotenuse of 2 belongs to the standard $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle. With both shorter sides equal to 1, Pythagoras gives $\\sqrt{2}$."},{"text":"

Two shorter sides: 1
Hypotenuse: $\\sqrt{3}$

","correct":false,"explanation":"This does not satisfy Pythagoras because $1^2+1^2=2$, not 3. The two shorter sides are 1 and the hypotenuse is $\\sqrt{2}$."},{"text":"

Two shorter sides: $1,\\sqrt{2}$
Hypotenuse: 2

","correct":false,"explanation":"This incorrectly uses unequal shorter sides, although a $45^\\circ$ right triangle is isosceles. Its standard side lengths are 1, 1 and $\\sqrt{2}$."}]},{"id":"dd78db56-be16-40d7-b56b-1e83187d2025","cardType":"basic","points":3,"front":"What are the three side lengths of the $30^\\circ$-$60^\\circ$-$90^\\circ$ special triangle?","back":"

Shortest side: 1
Middle side: $\\sqrt{3}$
Hypotenuse: 2

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

Shortest side: 1
Middle side: $\\sqrt{3}$
Hypotenuse: 2

","correct":true,"explanation":"Bisecting an equilateral triangle of side 2 produces a right triangle with hypotenuse 2, shortest side 1 and middle side $\\sqrt{2^2-1^2}=\\sqrt{3}$."},{"text":"

Shortest side: 1
Middle side: $\\sqrt{2}$
Hypotenuse: 2

","correct":false,"explanation":"The value $\\sqrt{2}$ comes from the $45^\\circ$ special triangle. Here Pythagoras gives the middle side as $\\sqrt{2^2-1^2}=\\sqrt{3}$."},{"text":"

Shortest side: 1
Middle side: 2
Hypotenuse: $\\sqrt{3}$

","correct":false,"explanation":"This reverses the roles of the hypotenuse and middle side. The hypotenuse must be longest, giving the ordered lengths 1, $\\sqrt{3}$ and 2."},{"text":"

Shortest side: 1
Middle side: 1
Hypotenuse: $\\sqrt{2}$

","correct":false,"explanation":"Making both perpendicular sides equal describes an isosceles right triangle, not a $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle. The correct lengths are 1, $\\sqrt{3}$ and 2."}]},{"id":"f17e0952-f444-4eb5-9dd9-e9b26cd3b565","cardType":"basic","points":2,"front":"What are the exact values of $\\sin(45^\\circ)$ and $\\cos(45^\\circ)$?","back":"They are equal: $\\frac{\\sqrt{2}}{2}$.","mcqOptions":[{"text":"They are equal: $\\frac{\\sqrt{2}}{2}$.","correct":true,"explanation":"In a $45^\\circ$ right triangle, the opposite and adjacent sides are equal. Dividing either leg, 1, by the hypotenuse, $\\sqrt{2}$, gives $\\frac{\\sqrt{2}}{2}$."},{"text":"They are equal: $\\frac{1}{2}$.","correct":false,"explanation":"The value $\\frac{1}{2}$ is $\\sin(30^\\circ)$, not either ratio at $45^\\circ$. Both $\\sin(45^\\circ)$ and $\\cos(45^\\circ)$ equal $\\frac{\\sqrt{2}}{2}$."},{"text":"They are equal: $\\frac{\\sqrt{3}}{2}$.","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{2}$ occurs for $\\cos(30^\\circ)$ and $\\sin(60^\\circ)$. At $45^\\circ$, both sine and cosine equal $\\frac{\\sqrt{2}}{2}$."},{"text":"They are equal: $\\sqrt{2}$.","correct":false,"explanation":"This misses the division by the hypotenuse: $1/\\sqrt{2}$ rationalises to $\\frac{\\sqrt{2}}{2}$, not $\\sqrt{2}$. Both exact values are $\\frac{\\sqrt{2}}{2}$."}]},{"id":"97044958-bda8-4830-9038-d945a7dd7cbe","cardType":"basic","points":1,"front":"What is the exact value of $\\sin(30^\\circ)$?","back":"$\\frac{1}{2}$","mcqOptions":[{"text":"$\\frac{1}{2}$","correct":true,"explanation":"In the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle, the side opposite $30^\\circ$ is 1 and the hypotenuse is 2, so $\\sin(30^\\circ)=\\frac{1}{2}$."},{"text":"$\\frac{\\sqrt{3}}{2}$","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{2}$ uses the longer adjacent side and is $\\cos(30^\\circ)$. Sine uses opposite over hypotenuse, giving $\\frac{1}{2}$."},{"text":"$\\frac{\\sqrt{2}}{2}$","correct":false,"explanation":"The value $\\frac{\\sqrt{2}}{2}$ belongs to sine and cosine at $45^\\circ$. For $30^\\circ$, opposite over hypotenuse is $1/2$."},{"text":"$\\frac{\\sqrt{3}}{3}$","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{3}$ is $\\tan(30^\\circ)$, formed from opposite over adjacent. Sine instead uses opposite over hypotenuse and equals $\\frac{1}{2}$."}]},{"id":"21d46f7f-12da-4050-a770-c7fa66b3a3af","cardType":"basic","points":1,"front":"What is the exact value of $\\cos(30^\\circ)$?","back":"$\\frac{\\sqrt{3}}{2}$","mcqOptions":[{"text":"$\\frac{\\sqrt{3}}{2}$","correct":true,"explanation":"In the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle, the side adjacent to $30^\\circ$ is $\\sqrt{3}$ and the hypotenuse is 2, giving $\\cos(30^\\circ)=\\frac{\\sqrt{3}}{2}$."},{"text":"$\\frac{1}{2}$","correct":false,"explanation":"The value $\\frac{1}{2}$ is obtained from the opposite side and is $\\sin(30^\\circ)$. Cosine uses adjacent over hypotenuse, giving $\\frac{\\sqrt{3}}{2}$."},{"text":"$\\frac{\\sqrt{2}}{2}$","correct":false,"explanation":"The value $\\frac{\\sqrt{2}}{2}$ belongs to cosine at $45^\\circ$. Using the $30^\\circ$ triangle’s adjacent side and hypotenuse gives $\\frac{\\sqrt{3}}{2}$."},{"text":"$\\frac{\\sqrt{3}}{3}$","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{3}$ is associated with $\\tan(30^\\circ)$, not cosine. Adjacent over hypotenuse gives the correct value $\\frac{\\sqrt{3}}{2}$."}]},{"id":"8cb6948e-d0c3-4a8f-bab2-75bbe49d920d","cardType":"basic","points":1,"front":"What is the exact value of $\\tan(30^\\circ)$?","back":"$\\frac{\\sqrt{3}}{3}$","mcqOptions":[{"text":"$\\frac{\\sqrt{3}}{3}$","correct":true,"explanation":"For $30^\\circ$, opposite over adjacent is $\\frac{1}{\\sqrt{3}}$. Rationalising the denominator by multiplying by $\\frac{\\sqrt{3}}{\\sqrt{3}}$ gives $\\frac{\\sqrt{3}}{3}$."},{"text":"$\\frac{1}{2}$","correct":false,"explanation":"The value $\\frac{1}{2}$ is $\\sin(30^\\circ)$ and uses the hypotenuse. Tangent uses opposite over adjacent, so its exact value is $\\frac{\\sqrt{3}}{3}$."},{"text":"$\\frac{\\sqrt{3}}{2}$","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{2}$ is $\\cos(30^\\circ)$ and includes the hypotenuse. Tangent is $1/\\sqrt{3}$, which rationalises to $\\frac{\\sqrt{3}}{3}$."},{"text":"$\\sqrt{3}$","correct":false,"explanation":"This reverses the tangent ratio by calculating adjacent over opposite, which is the cotangent. Opposite over adjacent gives $1/\\sqrt{3}=\\frac{\\sqrt{3}}{3}$."}]},{"id":"d7c2d3c6-c394-47d8-8d8d-2fc09d67eaf7","cardType":"basic","points":1,"front":"What is the exact value of $\\tan(45^\\circ)$?","back":"1","mcqOptions":[{"text":"1","correct":true,"explanation":"A $45^\\circ$ right triangle has equal opposite and adjacent sides. Their ratio is therefore $1/1=1$, so $\\tan(45^\\circ)=1$."},{"text":"$\\frac{\\sqrt{2}}{2}$","correct":false,"explanation":"The value $\\frac{\\sqrt{2}}{2}$ is $\\sin(45^\\circ)$ and $\\cos(45^\\circ)$ because each uses the hypotenuse. Tangent compares the equal legs and equals 1."},{"text":"$\\sqrt{2}$","correct":false,"explanation":"The value $\\sqrt{2}$ is the hypotenuse in the standard $45^\\circ$ triangle, not its tangent. Opposite and adjacent are both 1, so tangent equals 1."},{"text":"$\\frac{\\sqrt{3}}{3}$","correct":false,"explanation":"The value $\\frac{\\sqrt{3}}{3}$ is $\\tan(30^\\circ)$. At $45^\\circ$, the opposite and adjacent sides are equal, so their ratio is 1."}]},{"id":"befa9bce-3c67-4c73-a662-69fb51372313","cardType":"basic","points":1,"front":"What is the exact value of $\\sin(60^\\circ)$?","back":"$\\frac{\\sqrt{3}}{2}$","mcqOptions":[{"text":"$\\frac{\\sqrt{3}}{2}$","correct":true,"explanation":"The exact sine values follow $\\frac{\\sqrt{x}}{2}$ for $x=0,1,2,3,4$. At $60^\\circ$, this gives $\\sin(60^\\circ)=\\frac{\\sqrt{3}}{2}$."},{"text":"$\\frac{1}{2}$","correct":false,"explanation":"This confuses the sine values for $30^\\circ$ and $60^\\circ$. The correct exact value is $\\sin(60^\\circ)=\\frac{\\sqrt{3}}{2}$."},{"text":"$\\frac{\\sqrt{2}}{2}$","correct":false,"explanation":"This is the exact value of $\\sin(45^\\circ)$, not $\\sin(60^\\circ)$. For an angle of $60^\\circ$, the correct value is $\\frac{\\sqrt{3}}{2}$."},{"text":"$\\sqrt{3}$","correct":false,"explanation":"This is the exact value of $\\tan(60^\\circ)$ and is greater than 1, so it cannot be a sine value. The correct answer is $\\frac{\\sqrt{3}}{2}$."}]},{"id":"63cb1461-c5f9-47f3-b398-368340a127c3","cardType":"basic","points":1,"front":"What is the exact value of $\\cos(60^\\circ)$?","back":"$\\frac{1}{2}$","mcqOptions":[{"text":"$\\frac{1}{2}$","correct":true,"explanation":"The exact cosine values decrease from 1 to 0 between $0^\\circ$ and $90^\\circ$. At $60^\\circ$, $\\cos(60^\\circ)=\\frac{1}{2}$."},{"text":"$\\frac{\\sqrt{3}}{2}$","correct":false,"explanation":"This is $\\sin(60^\\circ)$, reflecting a common confusion between sine and cosine. The correct exact value is $\\cos(60^\\circ)=\\frac{1}{2}$."},{"text":"$\\frac{\\sqrt{2}}{2}$","correct":false,"explanation":"This is the exact cosine value at $45^\\circ$, not $60^\\circ$. The correct value at $60^\\circ$ is $\\frac{1}{2}$."},{"text":"$\\sqrt{3}$","correct":false,"explanation":"This is $\\tan(60^\\circ)$ and is greater than 1, whereas cosine cannot exceed 1. The correct exact value is $\\frac{1}{2}$."}]},{"id":"0cb8a9a7-016c-4fb9-8e32-b7245db55be3","cardType":"basic","points":1,"front":"What is the exact value of $\\tan(60^\\circ)$?","back":"$\\sqrt{3}$","mcqOptions":[{"text":"$\\sqrt{3}$","correct":true,"explanation":"Using $\\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}$ gives $\\tan(60^\\circ)=\\frac{\\sqrt{3}/2}{1/2}=\\sqrt{3}$."},{"text":"$\\frac{1}{\\sqrt{3}}$","correct":false,"explanation":"This is $\\tan(30^\\circ)$, produced by reversing the opposite and adjacent sides for $60^\\circ$. The correct exact value is $\\tan(60^\\circ)=\\sqrt{3}$."},{"text":"$\\frac{\\sqrt{3}}{2}$","correct":false,"explanation":"This is $\\sin(60^\\circ)$, not the tangent. Dividing sine by cosine gives $\\frac{\\sqrt{3}/2}{1/2}=\\sqrt{3}$."},{"text":"$1$","correct":false,"explanation":"This is the tangent value at $45^\\circ$, where opposite and adjacent sides are equal. At $60^\\circ$, the correct exact value is $\\sqrt{3}$."}]},{"id":"aebd50d0-7605-49ee-82e4-ecc35d00b56a","cardType":"cloze","points":null,"front":"At $90^\\circ$, $\\sin(90^\\circ) = \\text{[...]}$ and $\\cos(90^\\circ) = 0$.","back":"At $90^\\circ$, $\\sin(90^\\circ) = \\boxed{1}$ and $\\cos(90^\\circ) = 0$.","mcqOptions":[{"text":"$$1$","correct":true,"explanation":"Sine is opposite divided by hypotenuse. At $90^\\circ$, the opposite side reaches the full hypotenuse, so their ratio and therefore $\\sin(90^\\circ)$ equal 1."},{"text":"$$0$","correct":false,"explanation":"This confuses sine with cosine at $90^\\circ$. While $\\cos(90^\\circ)=0$, the sine value reaches its maximum, so $\\sin(90^\\circ)=1$."},{"text":"$$-1$","correct":false,"explanation":"A sine value of $-1$ occurs at $270^\\circ$, not $90^\\circ$. In the first quadrant sine is positive, and $\\sin(90^\\circ)=1$."},{"text":"$$\\frac{1}{2}$","correct":false,"explanation":"This is the exact sine value at $30^\\circ$, not $90^\\circ$. Sine increases to its maximum value of 1 at $90^\\circ$."}]},{"id":"af405b14-728a-421f-ba4b-06768a0e9e41","cardType":"cloze","points":null,"front":"At $90^\\circ$, $\\sin(90^\\circ) = 1$ and $\\cos(90^\\circ) = \\text{[...]}$.","back":"At $90^\\circ$, $\\sin(90^\\circ) = 1$ and $\\cos(90^\\circ) = \\boxed{0}$.","mcqOptions":[{"text":"$$0$","correct":true,"explanation":"Cosine is adjacent divided by hypotenuse. At $90^\\circ$, the adjacent horizontal component is zero, so $\\cos(90^\\circ)=0$."},{"text":"$$1$","correct":false,"explanation":"This is $\\cos(0^\\circ)$ and also $\\sin(90^\\circ)$, so it reflects an angle or function mix-up. The correct value is $\\cos(90^\\circ)=0$."},{"text":"$$-1$","correct":false,"explanation":"A cosine value of $-1$ occurs at $180^\\circ$, not $90^\\circ$. At $90^\\circ$, the horizontal component is zero, so the cosine is 0."},{"text":"$$\\frac{1}{2}$","correct":false,"explanation":"This is the exact value of $\\cos(60^\\circ)$, not $\\cos(90^\\circ)$. Cosine decreases to 0 as the angle reaches $90^\\circ$."}]},{"id":"927669da-2bcf-42b9-ac73-9a83bf97aac6","cardType":"basic","points":2,"front":"Why is $\\tan(90^\\circ)$ not defined?","back":"The adjacent side is zero, and dividing by zero is impossible.","mcqOptions":[{"text":"The adjacent side is zero, and dividing by zero is impossible.","correct":true,"explanation":"Tangent is opposite divided by adjacent. At $90^\\circ$ the adjacent side is zero, so calculating tangent would require division by zero, which is undefined."},{"text":"The opposite side is zero, and dividing by zero is impossible.","correct":false,"explanation":"This swaps the numerator and denominator in the tangent ratio. At $90^\\circ$ the adjacent side, not the opposite side, is zero, making division by zero unavoidable."},{"text":"The hypotenuse is zero, and dividing by zero is impossible.","correct":false,"explanation":"The hypotenuse cannot be zero in a non-degenerate right-angled triangle. Tangent is opposite divided by adjacent, and at $90^\\circ$ it is the adjacent side that is zero."},{"text":"The adjacent side is zero, and zero divided by a number is impossible.","correct":false,"explanation":"Zero divided by a non-zero number is defined and equals zero. The problem is that tangent requires the opposite side to be divided by the zero adjacent side."}]},{"id":"84eca0c1-450c-449d-877b-284d723439af","cardType":"cloze","points":null,"front":"The sine values from $0^\\circ$ to $90^\\circ$ follow the pattern $\\frac{\\sqrt{x}}{2}$, where $x$ values are $\\text{[...]}$.","back":"The sine values from $0^\\circ$ to $90^\\circ$ follow the pattern $\\frac{\\sqrt{x}}{2}$, where $x$ values are $\\boxed{0, 1, 2, 3, 4}$.","mcqOptions":[{"text":"$$0, 1, 2, 3, 4$","correct":true,"explanation":"For $0^\\circ$, $30^\\circ$, $45^\\circ$, $60^\\circ$ and $90^\\circ$, the sine values are formed by substituting $x=0,1,2,3,4$ into $\\frac{\\sqrt{x}}{2}$."},{"text":"$$0, 1, 4, 9, 16$","correct":false,"explanation":"These are square numbers, which would cancel the square root and produce values exceeding 1. The required $x$ values are $0,1,2,3,4$."},{"text":"$$4, 3, 2, 1, 0$","correct":false,"explanation":"This reverses the pattern and would make sine decrease from 1 to 0. Between $0^\\circ$ and $90^\\circ$, sine increases, using $x=0,1,2,3,4$."},{"text":"$$1, 2, 3, 4, 5$","correct":false,"explanation":"This off-by-one pattern incorrectly gives $\\sin(0^\\circ)=\\frac{1}{2}$ and makes the final value exceed 1. The correct sequence is $0,1,2,3,4$."}]},{"id":"0fe7ee57-85aa-42a6-a673-9fbd284f452a","cardType":"basic","points":3,"front":"In a triangle with a $30^\\circ$ angle and an opposite side of 18 cm, how do you find the hypotenuse $h$?","back":"

Use $\\sin(30^\\circ) = \\frac{18}{h}$
Substitute $\\frac{1}{2} = \\frac{18}{h}$
Solve for $h = 36$ cm

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

Use $\\sin(30^\\circ) = \\frac{18}{h}$
Substitute $\\frac{1}{2} = \\frac{18}{h}$
Solve for $h = 36$ cm

","correct":true,"explanation":"Sine is opposite divided by hypotenuse, so $\\sin(30^\\circ)=\\frac{18}{h}$. Substituting $\\frac{1}{2}$ and rearranging gives $h=18\\div\\frac{1}{2}=36$ cm."},{"text":"

Use $\\cos(30^\\circ) = \\frac{18}{h}$
Substitute $\\frac{\\sqrt{3}}{2} = \\frac{18}{h}$
Solve for $h = 12\\sqrt{3}$ cm

","correct":false,"explanation":"Cosine uses the adjacent side, but 18 cm is identified as the opposite side. Sine must be used, giving $\\frac{1}{2}=\\frac{18}{h}$ and $h=36$ cm."},{"text":"

Use $\\sin(30^\\circ) = \\frac{h}{18}$
Substitute $\\frac{1}{2} = \\frac{h}{18}$
Solve for $h = 9$ cm

","correct":false,"explanation":"This reverses the sine ratio: sine is opposite divided by hypotenuse, not hypotenuse divided by opposite. The equation should be $\\frac{1}{2}=\\frac{18}{h}$, giving $h=36$ cm."},{"text":"

Use $\\sin(60^\\circ) = \\frac{18}{h}$
Substitute $\\frac{\\sqrt{3}}{2} = \\frac{18}{h}$
Solve for $h = 12\\sqrt{3}$ cm

","correct":false,"explanation":"This uses the complementary angle instead of the stated $30^\\circ$ angle while still treating 18 cm as opposite. Using $\\sin(30^\\circ)=\\frac{18}{h}$ gives $h=36$ cm."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Exact trig values","title":"Exact trig values Flashcards","topbar":{"backHref":"/gcse/maths/edexcel/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"3974095b-6644-40bd-bbfe-559075707fa7","walled":"$L38"}],"$L39","$L3a"]