Trigonometry — Identifying sin, cos, tan — Part 2 — Dynamic Practice
Trigonometry — Identifying sin, cos, tan — Part 2 — Dynamic Practice. Practice questions to deepen understanding of identifying sine, cosine, and tangent in right triangles — advanced level. Online math practice with full solutions and detailed explanations.
Dynamic advanced practice in identifying trigonometric ratios in right triangles — opposite, adjacent, hypotenuse, plus mixed problems. New questions every attempt.
Question 1
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 60, the leg adjacent to the angle is 45, and the hypotenuse is 75.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 60
• Adjacent leg = 45
• Hypotenuse = 75
• Opposite leg = 60
• Adjacent leg = 45
• Hypotenuse = 75
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{60}{45}\)
Question 2
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 30, the leg adjacent to the angle is 72, and the hypotenuse is 78.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 30
• Adjacent leg = 72
• Hypotenuse = 78
• Opposite leg = 30
• Adjacent leg = 72
• Hypotenuse = 78
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{30}{72}\)
Question 3
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 36, the leg adjacent to the angle is 15, and the hypotenuse is 39.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 36
• Adjacent leg = 15
• Hypotenuse = 39
• Opposite leg = 36
• Adjacent leg = 15
• Hypotenuse = 39
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{15}{39}\)
Question 4
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 15, the leg adjacent to the angle is 8, and the hypotenuse is 17.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 15
• Adjacent leg = 8
• Hypotenuse = 17
• Opposite leg = 15
• Adjacent leg = 8
• Hypotenuse = 17
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{15}{8}\)
Question 5
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 108, the leg adjacent to the angle is 45, and the hypotenuse is 117.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 108
• Adjacent leg = 45
• Hypotenuse = 117
• Opposite leg = 108
• Adjacent leg = 45
• Hypotenuse = 117
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{108}{45}\)
Question 6
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 27, the leg adjacent to the angle is 120, and the hypotenuse is 123.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 27
• Adjacent leg = 120
• Hypotenuse = 123
• Opposite leg = 27
• Adjacent leg = 120
• Hypotenuse = 123
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{120}{123}\)
Question 7
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 60, the leg adjacent to the angle is 32, and the hypotenuse is 68.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 60
• Adjacent leg = 32
• Hypotenuse = 68
• Opposite leg = 60
• Adjacent leg = 32
• Hypotenuse = 68
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{60}{32}\)
Question 8
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 72, the leg adjacent to the angle is 30, and the hypotenuse is 78.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 72
• Adjacent leg = 30
• Hypotenuse = 78
• Opposite leg = 72
• Adjacent leg = 30
• Hypotenuse = 78
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{30}{78}\)
Question 9
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 48, the leg adjacent to the angle is 90, and the hypotenuse is 102.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 48
• Adjacent leg = 90
• Hypotenuse = 102
• Opposite leg = 48
• Adjacent leg = 90
• Hypotenuse = 102
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{48}{102}\)
Question 10
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 63, the leg adjacent to the angle is 60, and the hypotenuse is 87.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 63
• Adjacent leg = 60
• Hypotenuse = 87
• Opposite leg = 63
• Adjacent leg = 60
• Hypotenuse = 87
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{63}{60}\)
Question 11
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 105, the leg adjacent to the angle is 36, and the hypotenuse is 111.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 105
• Adjacent leg = 36
• Hypotenuse = 111
• Opposite leg = 105
• Adjacent leg = 36
• Hypotenuse = 111
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{36}{111}\)
Question 12
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 21, the leg adjacent to the angle is 72, and the hypotenuse is 75.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 21
• Adjacent leg = 72
• Hypotenuse = 75
• Opposite leg = 21
• Adjacent leg = 72
• Hypotenuse = 75
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{21}{72}\)
Question 13
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 15, the leg adjacent to the angle is 36, and the hypotenuse is 39.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 15
• Adjacent leg = 36
• Hypotenuse = 39
• Opposite leg = 15
• Adjacent leg = 36
• Hypotenuse = 39
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{36}{39}\)
Question 14
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 15, the leg adjacent to the angle is 20, and the hypotenuse is 25.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 15
• Adjacent leg = 20
• Hypotenuse = 25
• Opposite leg = 15
• Adjacent leg = 20
• Hypotenuse = 25
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{15}{20}\)
Question 15
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 9, the leg adjacent to the angle is 12, and the hypotenuse is 15.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 9
• Adjacent leg = 12
• Hypotenuse = 15
• Opposite leg = 9
• Adjacent leg = 12
• Hypotenuse = 15
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{9}{15}\)
Question 16
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 42, the leg adjacent to the angle is 40, and the hypotenuse is 58.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 42
• Adjacent leg = 40
• Hypotenuse = 58
• Opposite leg = 42
• Adjacent leg = 40
• Hypotenuse = 58
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{42}{58}\)
Question 17
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 24, the leg adjacent to the angle is 18, and the hypotenuse is 30.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 24
• Adjacent leg = 18
• Hypotenuse = 30
• Opposite leg = 24
• Adjacent leg = 18
• Hypotenuse = 30
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{24}{30}\)
Question 18
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 30, the leg adjacent to the angle is 16, and the hypotenuse is 34.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 30
• Adjacent leg = 16
• Hypotenuse = 34
• Opposite leg = 30
• Adjacent leg = 16
• Hypotenuse = 34
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{30}{34}\)
Question 19
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 32, the leg adjacent to the angle is 60, and the hypotenuse is 68.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 32
• Adjacent leg = 60
• Hypotenuse = 68
• Opposite leg = 32
• Adjacent leg = 60
• Hypotenuse = 68
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{32}{68}\)
Question 20
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 7, the leg adjacent to the angle is 24, and the hypotenuse is 25.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 7
• Adjacent leg = 24
• Hypotenuse = 25
• Opposite leg = 7
• Adjacent leg = 24
• Hypotenuse = 25
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{7}{25}\)
Question 21
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 24, the leg adjacent to the angle is 70, and the hypotenuse is 74.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 24
• Adjacent leg = 70
• Hypotenuse = 74
• Opposite leg = 24
• Adjacent leg = 70
• Hypotenuse = 74
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{24}{70}\)
Question 22
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 12, the leg adjacent to the angle is 16, and the hypotenuse is 20.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 12
• Adjacent leg = 16
• Hypotenuse = 20
• Opposite leg = 12
• Adjacent leg = 16
• Hypotenuse = 20
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{12}{20}\)
Question 23
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 20, the leg adjacent to the angle is 15, and the hypotenuse is 25.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 20
• Adjacent leg = 15
• Hypotenuse = 25
• Opposite leg = 20
• Adjacent leg = 15
• Hypotenuse = 25
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{15}{25}\)
Question 24
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 10, the leg adjacent to the angle is 24, and the hypotenuse is 26.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 10
• Adjacent leg = 24
• Hypotenuse = 26
• Opposite leg = 10
• Adjacent leg = 24
• Hypotenuse = 26
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{10}{24}\)
Question 25
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 9, the leg adjacent to the angle is 12, and the hypotenuse is 15.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 9
• Adjacent leg = 12
• Hypotenuse = 15
• Opposite leg = 9
• Adjacent leg = 12
• Hypotenuse = 15
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{9}{12}\)
Question 26
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 24, the leg adjacent to the angle is 45, and the hypotenuse is 51.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 24
• Adjacent leg = 45
• Hypotenuse = 51
• Opposite leg = 24
• Adjacent leg = 45
• Hypotenuse = 51
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{45}{51}\)
Question 27
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 24, the leg adjacent to the angle is 10, and the hypotenuse is 26.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 24
• Adjacent leg = 10
• Hypotenuse = 26
• Opposite leg = 24
• Adjacent leg = 10
• Hypotenuse = 26
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{24}{26}\)
Question 28
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 120, the leg adjacent to the angle is 27, and the hypotenuse is 123.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 120
• Adjacent leg = 27
• Hypotenuse = 123
• Opposite leg = 120
• Adjacent leg = 27
• Hypotenuse = 123
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{120}{27}\)
Question 29
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 40, the leg adjacent to the angle is 42, and the hypotenuse is 58.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 40
• Adjacent leg = 42
• Hypotenuse = 58
• Opposite leg = 40
• Adjacent leg = 42
• Hypotenuse = 58
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{40}{58}\)
Question 30
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 15, the leg adjacent to the angle is 8, and the hypotenuse is 17.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 15
• Adjacent leg = 8
• Hypotenuse = 17
• Opposite leg = 15
• Adjacent leg = 8
• Hypotenuse = 17
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{15}{17}\)
Question 31
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 20, the leg adjacent to the angle is 21, and the hypotenuse is 29.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 20
• Adjacent leg = 21
• Hypotenuse = 29
• Opposite leg = 20
• Adjacent leg = 21
• Hypotenuse = 29
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{20}{29}\)
Question 32
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 3, the leg adjacent to the angle is 4, and the hypotenuse is 5.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 3
• Adjacent leg = 4
• Hypotenuse = 5
• Opposite leg = 3
• Adjacent leg = 4
• Hypotenuse = 5
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{3}{4}\)
Question 33
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 30, the leg adjacent to the angle is 72, and the hypotenuse is 78.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 30
• Adjacent leg = 72
• Hypotenuse = 78
• Opposite leg = 30
• Adjacent leg = 72
• Hypotenuse = 78
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{72}{78}\)
Question 34
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 35, the leg adjacent to the angle is 12, and the hypotenuse is 37.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 35
• Adjacent leg = 12
• Hypotenuse = 37
• Opposite leg = 35
• Adjacent leg = 12
• Hypotenuse = 37
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{35}{12}\)
Question 35
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 48, the leg adjacent to the angle is 20, and the hypotenuse is 52.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 48
• Adjacent leg = 20
• Hypotenuse = 52
• Opposite leg = 48
• Adjacent leg = 20
• Hypotenuse = 52
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{48}{20}\)
Question 36
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 10, the leg adjacent to the angle is 24, and the hypotenuse is 26.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 10
• Adjacent leg = 24
• Hypotenuse = 26
• Opposite leg = 10
• Adjacent leg = 24
• Hypotenuse = 26
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{10}{26}\)
Question 37
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 24, the leg adjacent to the angle is 45, and the hypotenuse is 51.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 24
• Adjacent leg = 45
• Hypotenuse = 51
• Opposite leg = 24
• Adjacent leg = 45
• Hypotenuse = 51
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{24}{45}\)
Question 38
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 72, the leg adjacent to the angle is 21, and the hypotenuse is 75.
What is \(\sin(\alpha)\)?
What is \(\sin(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 72
• Adjacent leg = 21
• Hypotenuse = 75
• Opposite leg = 72
• Adjacent leg = 21
• Hypotenuse = 75
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
Answer: \(\frac{72}{75}\)
Question 39
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 6, the leg adjacent to the angle is 8, and the hypotenuse is 10.
What is \(\cos(\alpha)\)?
What is \(\cos(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 6
• Adjacent leg = 8
• Hypotenuse = 10
• Opposite leg = 6
• Adjacent leg = 8
• Hypotenuse = 10
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Answer: \(\frac{8}{10}\)
Question 40
2.50 pts
📐 In a right triangle, the leg opposite to angle a is 40, the leg adjacent to the angle is 30, and the hypotenuse is 50.
What is \(\tan(\alpha)\)?
What is \(\tan(\alpha)\)?
Explanation:
Solution - Identifying a Trigonometric Function:
📝 The rules:
\(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
\(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
🔢 In our case:
• Opposite leg = 40
• Adjacent leg = 30
• Hypotenuse = 50
• Opposite leg = 40
• Adjacent leg = 30
• Hypotenuse = 50
\(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\)
Answer: \(\frac{40}{30}\)