Solving Right Triangles: Pythagorean Theorem, Trigonometry - Precalculus

In right triangle $\triangle ABC$ , $\angle C = 90^\circ$ and $AB$ is the hypotenuse. If $\angle A = 30^\circ$ and $AB = 14$ , what is the exact length of side $BC$ ?
Answer: $7$ This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle A = 30°, the side BC is the opposite side and AB is the hypotenuse, so we use sin(30°) = BC/AB, which gives 1/2 = BC/14, solving to get BC = 14 × (1/2) = 7. Choice A is correct because sin(30°) = 1/2, so BC = AB × sin(30°) = 14 × (1/2) = 7. Choice B incorrectly uses cos(30°) = √3/2 instead of sin(30°), giving BC = 14 × (√3/2) = 7√3, which would be the length of AC, not BC. This is a 30-60-90 triangle scaled by factor 7, where the sides are in ratio 1:√3:2, confirming BC = 7. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.
If $\cos(2x + 10°) = \sin(3x - 20°)$ and both angles are acute, what is the value of $x$ ?
Answer: $x = 20°$ because $\cos(\theta) = \sin(90° - \theta)$ requires $2x + 10° = 90° - (3x - 20°)$ When you encounter an equation where cosine equals sine, you need to use the cofunction identity: $\cos(\theta) = \sin(90° - \theta)$ . This relationship tells us that cosine of an angle equals sine of its complement. Given $\cos(2x + 10°) = \sin(3x - 20°)$ , we can rewrite the left side using the cofunction identity: $\sin(90° - (2x + 10°)) = \sin(3x - 20°)$ . This simplifies to $\sin(80° - 2x) = \sin(3x - 20°)$ . Since both expressions equal the same sine value and we're told both angles are acute, the angles inside the sine functions must be equal: $80° - 2x = 3x - 20°$ . Solving: $80° + 20° = 3x + 2x$ , so $100° = 5x$ , giving us $x = 20°$ . Now examining the answer choices: Choice A gets the correct answer $x = 20°$ but uses incorrect reasoning by adding the angles together rather than setting up the proper cofunction relationship. Choice B applies the cofunction identity correctly with $2x + 10° = 90° - (3x - 20°)$ , but makes an algebraic error, getting $x = 14°$ . Choice C incorrectly adds $90°$ to the second angle instead of subtracting it, leading to $x = 30°$ . Choice D correctly applies the cofunction identity as $2x + 10° = 90° - (3x - 20°)$ and solves accurately to get $x = 20°$ . Remember: when cosine equals sine, use the cofunction identity $\cos(\theta) = \sin(90° - \theta)$ and be careful with your algebra when distributing the negative sign.
In right triangle $MNO$ , the right angle is at $N$ . If $MN=12$ and the hypotenuse $MO=13$ , what is the value of $\cos(\angle M)$ ?
Answer: $\frac{12}{13}$ This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ , and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ . From the perspective of angle M, the side of length 12 is the adjacent, and the hypotenuse is 13, so we use $\cos(M) = \frac{12}{13}$ . Choice B is correct because it uses the cosine ratio with adjacent side 12 and hypotenuse 13, yielding $\frac{12}{13}$ . Choice A inverts the ratio, calculating hypotenuse/adjacent = $\frac{13}{12}$ instead of adjacent/hypotenuse. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, $\sin(\theta) = \cos(90^\circ - \theta)$ , which explains why the sine of one acute angle equals the cosine of the other.
In right triangle $DEF$ , the right angle is at $E$ . The hypotenuse is $DF = 10$ and one leg is $DE = 6$ . What is the length of the other leg $EF$ ? Give your answer in simplest form.
Answer: $8$ This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given the hypotenuse of length 10 and one leg of length 6, we rearrange the Pythagorean theorem: b² = 10² - 6², which gives 100 - 36 = b², so b² = 64, and taking the square root yields b = 8. Choice B is correct because it shows correct substitution into the Pythagorean theorem, identifying the hypotenuse as 10 and subtracting the square of the known leg 6 to find the other leg. Choice C incorrectly adds the squares instead of subtracting: √(100 + 36) = √136 instead of √(100 - 36) = 8. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.
In right triangle $PQR$ with right angle at $Q$ , $PQ=5$ and $QR=12$ . Which equation can be used to find the hypotenuse $PR$ ?
Answer: $PR^2=5^2+12^2$ This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given legs of length 5 and 12, we substitute into the Pythagorean theorem: 5² + 12² = PR² to find the hypotenuse. Choice A is correct because it shows correct substitution into the Pythagorean theorem with 5 and 12 as legs, setting up PR² = 5² + 12². Choice D incorrectly subtracts the squares, using PR² = 5² - 12², which would apply if mistakenly treating a leg as hypotenuse. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.
In right triangle $ABC$ , the right angle is at $C$ . If $AC=12$ and $BC=5$ , what is the value of $\tan(\angle A)$ ?
Answer: $\dfrac{5}{12}$ This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle A, the side BC = 5 is opposite to angle A, and the side AC = 12 is adjacent to angle A, so we use tan(A) = opposite/adjacent = 5/12. Choice C is correct because tan(A) = BC/AC = 5/12, correctly identifying BC as opposite and AC as adjacent to angle A. Choice A reverses the ratio, calculating 12/5 instead of 5/12, which inverts the tangent ratio by putting adjacent over opposite. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. When using tangent, you don't need the hypotenuse—just the two legs, making it useful when the hypotenuse is unknown.
In right triangle $ABC$ , the right angle is at $C$ . The legs are $AC = 12$ and $BC = 5$ , and the hypotenuse is $AB = c$ . What is the exact length of side $AB$ ?
Answer: 13 This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given sides of length 12 and 5, we substitute into the Pythagorean theorem: 12² + 5² = c², which gives 144 + 25 = c², so c² = 169, and taking the square root yields c = 13. Choice A is correct because it shows correct substitution into the Pythagorean theorem with the legs 12 and 5 from the triangle, resulting in the hypotenuse of 13. Choice B makes the error of adding the sides instead of adding their squares: 12 + 5 = 17 instead of √(12² + 5²) = 13. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.
In right triangle $ABC$ with a right angle at $C$ , $AC = 6$ and $BC = 8$ . Which equation can be used to find the hypotenuse $AB$ ?
Answer: $6^2 + 8^2 = AB^2$ This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given sides of length 6 and 8, we substitute into the Pythagorean theorem: 6² + 8² = AB², which gives 36 + 64 = AB², so AB² = 100, and taking the square root yields AB = 10. Choice C is correct because it shows correct substitution into the Pythagorean theorem, identifying the legs as 6 and 8 and the hypotenuse as AB in a² + b² = c². Choice A makes the error of adding the sides instead of adding their squares: 6 + 8 = AB instead of √(6² + 8²). Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.
In right triangle $MNO$ , the right angle is at $N$ . The hypotenuse is $MO = 13$ , and $MN = 5$ . What is the length of $NO$ ?
Answer: $12$ This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This is a 5-12-13 Pythagorean triple, so we can recognize immediately that the missing side is 12 without calculation. Choice C is correct because it shows correct substitution into the Pythagorean theorem, identifying the hypotenuse as 13 and subtracting the square of the known leg 5 to find the other leg as 12. Choice D incorrectly adds the squares instead of subtracting: √(169 + 25) = √194 instead of √(169 - 25) = 12. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.
In triangle $ABC$ shown, angle $C = 90¬∞$ , $AB = 26$ , and $BC = 10$ . If $\sin(A) = \cos(B)$ , what is the relationship between angles $A$ and $B$ , and what is the measure of angle $A$ ?
Answer: Angles $A$ and $B$ are complementary, and $\angle A = \arcsin\left(\frac{10}{26}\right) \approx 22.6¬∞$ In any right triangle, the two acute angles are complementary (sum to 90¬∞). The given condition sin(A) = cos(B) confirms this relationship since cos(B) = sin(90¬∞ - B) = sin(A), which means A = 90¬∞ - B, or A + B = 90¬∞. First, find AC using the Pythagorean theorem: AC = ‚àö(26¬≤ - 10¬≤) = ‚àö(676 - 100) = ‚àö576 = 24. For angle A, sin(A) = opposite/hypotenuse = BC/AB = 10/26. Therefore, ‚à†A = arcsin(10/26) ‚âà 22.6¬∞. Choice B uses arccosine instead of arcsine. Choice C incorrectly states the angles are supplementary and uses the wrong ratio. Choice D incorrectly states the angles are equal rather than complementary.
In right triangle $PQR$ , the right angle is at $Q$ . If $PQ = 12$ and $PR = 13$ (the hypotenuse), what is the exact value of $\sin(\angle P)$ ?
Answer: $\frac{5}{13}$ This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle P, the side of length 12 is the adjacent, and the hypotenuse is 13, so we first find the opposite side as √(13² - 12²) = 5, then use sin(P) = opposite/hypotenuse = 5/13. Choice B is correct because it uses the sine ratio with opposite side 5 and hypotenuse 13 from the perspective of angle P. Choice A reverses the sine and cosine ratios, using cos instead of sin, which swaps the opposite and adjacent sides. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.
A ladder is 13 ft long and leans against a vertical wall. The bottom of the ladder is 5 ft from the wall, forming a right triangle with the ground. What is the height on the wall that the ladder reaches?
Answer: $12$ ft This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The ladder forms a right triangle where the ladder itself is the hypotenuse (13 ft), the distance from the wall is one leg (5 ft), and the height on the wall is the other leg, so we use: 5² + h² = 13², which gives 25 + h² = 169, so h² = 144, and taking the square root yields h = 12. Choice B is correct because this is a 5-12-13 Pythagorean triple, so we can recognize immediately that the missing side is 12 without calculation. Choice D incorrectly adds the squares: √(13² + 5²) = √(169 + 25) = √194, treating both given measurements as legs instead of recognizing that 13 is the hypotenuse. Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation. In real-world problems, identify which measurement represents the hypotenuse (often a ladder, ramp, or diagonal distance) versus the legs (usually horizontal and vertical distances).
In right triangle $XYZ$ , the right angle is at $Y$ . The hypotenuse is $XZ = 13$ and one leg is $YZ = 12$ . What is the exact length of side $XY$ ?
Answer: 5 This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given the hypotenuse of length 13 and one leg of 12, we rearrange the Pythagorean theorem to find the other leg a: a² + 12² = 13², which gives a² + 144 = 169, so a² = 25, and taking the square root yields a = 5. Choice B is correct because it shows correct substitution into the Pythagorean theorem with the hypotenuse 13 and leg 12, resulting in the other leg of 5. Choice D incorrectly treats 12 as the hypotenuse, using 12² = 13² - a² or similar misidentification. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.
In right triangle $ABC$ , the right angle is at $C$ . If $\angle A=60^\circ$ and $AC=8$ (adjacent to $\angle A$ ), which equation can be used to find the hypotenuse $AB$ ?
Answer: $\cos(60^\circ)=\dfrac{8}{AB}$ This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle A = 60°, the side AC = 8 is adjacent to angle A, and AB is the hypotenuse we need to find, so we use cos(60°) = adjacent/hypotenuse = 8/AB. Choice B is correct because cos(60°) = AC/AB = 8/AB, correctly identifying AC as the adjacent side to angle A and AB as the hypotenuse. Choice A incorrectly uses sine instead of cosine, which would require the opposite side BC rather than the adjacent side AC that we're given. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. When setting up equations to solve for an unknown side, place the unknown in the denominator if it's the hypotenuse or larger side, then cross-multiply to solve.
A surveyor stands $24$ m from the base of a vertical tower. The angle of elevation from the surveyor to the top of the tower is $45^\circ$ . Using the given information, what is the height of the tower?
Answer: $24$ m This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Since we know the adjacent side (24 m distance to base) and the angle of elevation (45°), and need the opposite side (height of tower), we use tan(45°) = height/24; since tan(45°)=1, height=24 × 1=24. Choice B is correct because it uses the tangent ratio with the given angle and adjacent side: 24 tan(45°) = 24 × 1 = 24 m. Choice C confuses the 45-45-90 triangle ratios, multiplying by √2 as if finding the hypotenuse instead of the opposite side. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known).
In right triangle $\triangle ABC$ , $\angle C = 90^\circ$ and $AB$ is the hypotenuse. If $AB = 13$ and $AC = 5$ , what is the exact length of leg $BC$ ?
Answer: $12$ This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Since we know the hypotenuse AB = 13 and one leg AC = 5, we need to find the other leg BC, so we rearrange: AC² + BC² = AB², which gives 5² + BC² = 13², so 25 + BC² = 169, therefore BC² = 144, and taking the square root yields BC = 12. Choice B is correct because when we substitute into the rearranged Pythagorean theorem, we get BC = √(13² - 5²) = √(169 - 25) = √144 = 12. Choice C incorrectly adds the squares instead of subtracting: √(169 + 25) = √194 instead of √(169 - 25) = 12. This is a 5-12-13 Pythagorean triple, so we can recognize immediately that the missing side is 12 without calculation. When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.
In right triangle $\triangle MNO$ , $\angle O = 90^\circ$ , the hypotenuse is $MN = 12$ cm, and $\angle M = 30^\circ$ . Using the given information, what is the exact length of side $NO$ (the side opposite $\angle M$ )?
Answer: $6$ cm This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle M (30°), the side NO is opposite, and MN=12 is the hypotenuse, so we use sin(M) = NO/12; since sin(30°)=1/2, NO=12 × (1/2)=6. Choice A is correct because it uses the sine ratio for the 30° angle: 12 sin(30°) = 12 × (1/2) = 6 cm. Choice B confuses the 30-60-90 triangle ratios with 45-45-90 triangle ratios, using √3 where it's not needed for the side opposite 30°. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.
A ramp rises $3$ m vertically over a horizontal run of $4$ m, forming a right triangle. Using the given information, what is the exact length (in m) of the ramp (the hypotenuse)? Give your answer in simplest form.
Answer: 5 This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This is a 3-4-5 Pythagorean triple, so we can recognize immediately that the missing side is 5 without calculation. Choice A is correct because it shows correct substitution into the Pythagorean theorem with legs 3 and 4, resulting in hypotenuse √(9 + 16) = √25 = 5. Choice B calculates √(3² + 4²) as √3² + √4² = 3 + 2 = 5, but actually errs in method while getting the number right; a true distractor might add instead, like 3 + 4 = 7. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.
A surveyor stands 15 m from the base of a tower on level ground. The angle of elevation to the top of the tower is $45^\circ$ . What is the height of the tower?
Answer: $15$ m This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the surveyor's position, the horizontal distance (15 m) is adjacent to the angle of elevation, and the tower height is opposite, so we use tan(45°) = height/15, solving to get height = 15 × tan(45°) = 15 × 1 = 15. Choice A is correct because tan(45°) = 1, so height = 15 × tan(45°) = 15 × 1 = 15 m. Choice B incorrectly treats this as a 45-45-90 triangle where the hypotenuse would be 15√2, but 15 m is the adjacent side (horizontal distance), not the hypotenuse. In a 45-45-90 triangle, the two legs are equal, so when the angle of elevation is 45°, the height equals the horizontal distance. Remember that for angle of elevation problems, use tangent when you know the horizontal distance and need the height, since tan(elevation angle) = height/horizontal distance.
A ladder is $10$ ft long and leans against a vertical wall, forming a $60^\circ$ angle with the ground. Using the given information, what is the exact height (in feet) the ladder reaches on the wall?
Answer: $5\sqrt{3}$ ft This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Since we know the hypotenuse (ladder length 10 ft) and need to find the opposite side to the 60° angle (height on the wall), we use the sine ratio: sin(60°) = height/10, so height = 10 × (√3/2) = 5√3. Choice C is correct because it uses the sine ratio with the given angle and hypotenuse: 10 sin(60°) = 10 × (√3/2) = 5√3 ft. Choice D confuses the 30-60-90 triangle ratios, using the longer leg (hypotenuse × √3/2) as if it were opposite 30° instead of 60°. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.

Start this quiz