Practice
Trigonometric Ratios of Acute Angles Practice
Fifty original questions on the six ratios, missing sides, cofunctions, identities, and right-triangle applications.
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Question 1
Explanation
Sine compares the opposite leg with the hypotenuse.
- Reasoning
Mark the selected angle, then identify the side that does not touch it.
Question 2
Explanation
Cosine compares the adjacent leg with the hypotenuse.
- Reasoning
Find the non-hypotenuse side that touches the selected angle.
Question 3
Explanation
Tangent compares the two legs: opposite over adjacent.
- Reasoning
Use TOA after naming the two legs relative to the angle.
Question 4
Explanation
Cosecant and sine are reciprocal partners.
- Reasoning
Pair sine-cosecant, cosine-secant, and tangent-cotangent.
Question 5
Explanation
Secant equals one divided by cosine.
- Reasoning
Recall the cosine-secant reciprocal pair.
Question 6
Explanation
Cotangent equals one divided by tangent.
- Reasoning
Recall the tangent-cotangent reciprocal pair.
Question 7
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{15}{17}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 8
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac8{17}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 9
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{15}8\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 10
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{25}{24}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 11
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{25}7\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 12
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac7{24}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 13
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{20}{29}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 14
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{21}{29}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 15
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{20}{21}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 16
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{41}{40}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 17
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac{41}9\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 18
Explanation
Relative to the selected angle, use the appropriate opposite, adjacent, and hypotenuse side lengths; the exact ratio is \(\frac9{40}\).
- Reasoning
Name the three sides from the selected angle before choosing a ratio.
Question 19
Explanation
The missing side satisfies \(6^2+b^2=10^2\), giving \(b=8\). The requested exact ratio is \(\frac45\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 20
Explanation
The missing side satisfies \(9^2+b^2=15^2\), giving \(b=12\). The requested exact ratio is \(\frac35\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 21
Explanation
The missing side satisfies \(12^2+b^2=13^2\), giving \(b=5\). The requested exact ratio is \(\frac5{12}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 22
Explanation
The missing side satisfies \(20^2+b^2=29^2\), giving \(b=21\). The requested exact ratio is \(\frac{29}{20}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 23
Explanation
The missing side satisfies \(28^2+b^2=53^2\), giving \(b=45\). The requested exact ratio is \(\frac{53}{45}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 24
Explanation
The missing side satisfies \(33^2+b^2=65^2\), giving \(b=56\). The requested exact ratio is \(\frac{33}{56}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 25
Explanation
The missing side satisfies \(48^2+b^2=73^2\), giving \(b=55\). The requested exact ratio is \(\frac{55}{73}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 26
Explanation
The missing side satisfies \(60^2+b^2=61^2\), giving \(b=11\). The requested exact ratio is \(\frac{60}{61}\).
- Reasoning
Use the Pythagorean theorem first, then label opposite and adjacent.
Question 27
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\sin 28^\circ=\cos 62^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 28
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\cos 17^\circ=\sin 73^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 29
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\tan 36^\circ=\cot 54^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 30
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\sec 41^\circ=\csc 49^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 31
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\csc 22^\circ=\sec 68^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 32
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\cot 13^\circ=\tan 77^\circ\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 33
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\sin(90^\circ-x)=\cos x\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 34
Explanation
The angles sum to \(90^\circ\), and the paired cofunctions exchange names, so \(\tan(90^\circ-y)=\cot y\).
- Reasoning
Check that the two angles are complementary, then use the correct cofunction pair.
Question 35
Explanation
Use a 5-12-13 triangle. Therefore the requested exact ratio is \(\frac{12}{13}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 36
Explanation
Use a 8-15-17 triangle. Therefore the requested exact ratio is \(\frac8{15}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 37
Explanation
Use legs 7 and 24, giving hypotenuse 25. Therefore the requested exact ratio is \(\frac7{25}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 38
Explanation
Secant gives adjacent 9 and hypotenuse 41; the other leg is 40. Therefore the requested exact ratio is \(\frac{40}{41}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 39
Explanation
Cosecant gives hypotenuse 29 and opposite 21; adjacent is 20. Therefore the requested exact ratio is \(\frac{20}{21}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 40
Explanation
Opposite 20 and hypotenuse 29 imply adjacent 21. Therefore the requested exact ratio is \(\frac{20}{21}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 41
Explanation
Adjacent 12 and hypotenuse 37 imply opposite 35. Therefore the requested exact ratio is \(\frac{37}{35}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 42
Explanation
Adjacent 11 and opposite 60 imply hypotenuse 61. Therefore the requested exact ratio is \(\frac{61}{11}\).
- Reasoning
Turn the given ratio into side labels, find the third side, and form the requested ratio.
Question 43
Explanation
The altitude bisects the base into 5 and 5; each right triangle is 5-12-13, so sine of the half-angle is 5/13.
- Reasoning
Split the base before forming the ratio.
Question 44
Explanation
A 3-4-5 triangle gives tangent 3/4 and cotangent 4/3; their sum is 25/12.
- Reasoning
Recover the full triangle, then combine exact fractions.
Question 45
Explanation
Cofunction gives \(\sin(90^\circ-x)=\cos x=5/13\). The 5-12-13 triangle gives \(\tan x=12/5\), and \(5/13+12/5=181/65\).
- Reasoning
Use the cofunction first, then the recovered triangle.
Question 46
Explanation
Legs 12 and 35 give hypotenuse 37. Thus sin²−cos²=(144−1225)/1369=−1081/1369.
- Reasoning
Build the Pythagorean triple before squaring ratios.
Question 47
Explanation
The altitude creates an 8-15-17 triangle. At a base angle, opposite is 15 and adjacent is 8.
- Reasoning
Halve the base and identify the selected base angle.
Question 48
Explanation
Because \(\csc B=\sec A=25/24\). A 7-24-25 triangle gives \(\tan A=7/24\), so the sum is \(32/24=4/3\).
- Reasoning
Use the cofunction relationship before combining ratios.
Question 49
Explanation
Opposite 16 and hypotenuse 65 give adjacent 63. Then sec=65/63 and cot=63/16, whose product is 65/16.
- Reasoning
Look for cancellation after recovering the missing side.
Question 50
Explanation
The horizontal leg is 7. Sine is 24/25 and cosine is 7/25, so (24/25)/(32/25)=3/4.
- Reasoning
Complete the right triangle, then simplify the compound expression.
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Questions to review
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- Question 1Sine definitionEasy
- Question 2Cosine definitionEasy
- Question 3Tangent definitionEasy
- Question 4Reciprocal functionsEasy
- Question 5Reciprocal functionsEasy
- Question 6Reciprocal functionsEasy
- Question 7Diagram-based trigonometric ratiosEasy
- Question 8Diagram-based trigonometric ratiosEasy
- Question 9Diagram-based trigonometric ratiosEasy
- Question 10Diagram-based trigonometric ratiosEasy
- Question 11Diagram-based trigonometric ratiosEasy
- Question 12Diagram-based trigonometric ratiosEasy
- Question 13Diagram-based trigonometric ratiosEasy
- Question 14Diagram-based trigonometric ratiosEasy
- Question 15Diagram-based trigonometric ratiosEasy
- Question 16Diagram-based trigonometric ratiosMedium
- Question 17Diagram-based trigonometric ratiosMedium
- Question 18Diagram-based trigonometric ratiosMedium
- Question 19Pythagorean trig ratiosMedium
- Question 20Pythagorean trig ratiosMedium
- Question 21Pythagorean trig ratiosMedium
- Question 22Pythagorean trig ratiosMedium
- Question 23Pythagorean trig ratiosMedium
- Question 24Pythagorean trig ratiosMedium
- Question 25Pythagorean trig ratiosMedium
- Question 26Pythagorean trig ratiosMedium
- Question 27Complementary-angle theoremMedium
- Question 28Complementary-angle theoremMedium
- Question 29Complementary-angle theoremMedium
- Question 30Complementary-angle theoremMedium
- Question 31Complementary-angle theoremMedium
- Question 32Complementary-angle theoremMedium
- Question 33Complementary-angle theoremMedium
- Question 34Complementary-angle theoremMedium
- Question 35Recovering ratios from one valueMedium
- Question 36Recovering ratios from one valueMedium
- Question 37Recovering ratios from one valueMedium
- Question 38Recovering ratios from one valueMedium
- Question 39Recovering ratios from one valueMedium
- Question 40Recovering ratios from one valueMedium
- Question 41Recovering ratios from one valueHard
- Question 42Recovering ratios from one valueHard
- Question 43Isosceles triangle decompositionHard
- Question 44Combining trig ratiosHard
- Question 45Cofunction identity synthesisHard
- Question 46Trig identities with exact fractionsHard
- Question 47Isosceles triangle decompositionHard
- Question 48Cofunction synthesisHard
- Question 49Reciprocal-ratio synthesisHard
- Question 50Right-triangle modelingHard