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Questions and Answers
What is the angle formed by a horizontal line and a line of sight to a point above?
What is the angle formed by a horizontal line and a line of sight to a point above?
Angle of Elevation
What is the positive number x such that a/x = x/b for positive numbers a and b?
What is the positive number x such that a/x = x/b for positive numbers a and b?
Geometric Mean
What is the angle formed by a horizontal line and a line of sight to a point below?
What is the angle formed by a horizontal line and a line of sight to a point below?
Angle of Depression
In a right triangle, what is the ratio of the length of the leg opposite the angle to the length of the hypotenuse?
In a right triangle, what is the ratio of the length of the leg opposite the angle to the length of the hypotenuse?
In a right triangle, what is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse?
In a right triangle, what is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse?
In a right triangle, what is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle?
In a right triangle, what is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle?
What do you call a ratio of two sides of a right triangle?
What do you call a ratio of two sides of a right triangle?
What is the form of a vector that lists the vertical and horizontal change from the initial point to the terminal point?
What is the form of a vector that lists the vertical and horizontal change from the initial point to the terminal point?
What is a quantity that has both magnitude and direction?
What is a quantity that has both magnitude and direction?
What do you call two vectors that have the same magnitude and the same direction?
What do you call two vectors that have the same magnitude and the same direction?
What are vectors with the same or opposite direction called?
What are vectors with the same or opposite direction called?
What is the length of a vector, written AB or V?
What is the length of a vector, written AB or V?
What is the vector that represents the sum of two given vectors?
What is the vector that represents the sum of two given vectors?
What term describes the orientation of a vector, determined by the angle the vector makes with a horizontal line?
What term describes the orientation of a vector, determined by the angle the vector makes with a horizontal line?
What is the geometric mean of the pair of numbers 5 and 20?
What is the geometric mean of the pair of numbers 5 and 20?
If a ramp for loading motorcycles onto a trailer is made at a 20-degree angle with the trailer 3.6 ft off the ground, what is the length of the ramp?
If a ramp for loading motorcycles onto a trailer is made at a 20-degree angle with the trailer 3.6 ft off the ground, what is the length of the ramp?
A pilot flying at an altitude of 1.8 km has an angle of depression of 31 degrees to the beginning of the runway and 23 degrees to the end of the runway. What is the length of the runway?
A pilot flying at an altitude of 1.8 km has an angle of depression of 31 degrees to the beginning of the runway and 23 degrees to the end of the runway. What is the length of the runway?
If a dam is 200 ft tall and leans at an 80-degree angle, with the base of the supporting beam placed 75 feet from the dam, how long must the beam be?
If a dam is 200 ft tall and leans at an 80-degree angle, with the base of the supporting beam placed 75 feet from the dam, how long must the beam be?
What is the magnitude of the vector (-5, 4)?
What is the magnitude of the vector (-5, 4)?
What is the direction of the vector (5, 2)?
What is the direction of the vector (5, 2)?
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Study Notes
Angles and Means
- Angle of Elevation: Formed by a horizontal line with a line of sight to a point above.
- Angle of Depression: Formed by a horizontal line with a line of sight to a point below.
- Geometric Mean: For positive numbers a and b, the positive number x such that a/x = x/b; for example, the geometric mean of 5 and 20 is 10.
Trigonometric Ratios
- Sine: In a right triangle, ratio of the leg opposite the angle to the hypotenuse.
- Cosine: In a right triangle, ratio of the leg adjacent to the angle to the hypotenuse.
- Tangent: In a right triangle, ratio of the leg opposite the angle to the leg adjacent to the angle.
- Trigonometric Ratio: Any ratio of two sides in a right triangle.
Vectors
- Vector: A quantity with both magnitude and direction.
- Component Form: The representation of a vector that lists vertical and horizontal changes from the initial to terminal point.
- Magnitude: The length of a vector, abbreviated as AB or V.
- Equal Vectors: Two vectors that are identical in both magnitude and direction.
- Parallel Vectors: Vectors that share the same or opposite direction.
- Resultant Vectors: The vector that represents the sum of two given vectors.
- Direction: The orientation of a vector, determined by the angle it makes with a horizontal line.
Practical Applications
- Ramp Design: To construct a ramp for motorcycles at a height of 3.6 ft and an angle of 20 degrees, the ramp's length should measure 10.53 ft.
- Altitude and Angles: A pilot at an altitude of 1.8 km with angles of depression of 31 degrees and 23 degrees to a runway finds its length to be 1.2 km.
- Supporting Beam: For a dam at an 80-degree angle and 200 ft tall, with a supporting beam base 75 ft away, the length required for the beam is 201 ft.
Conclusion
- Vector (-5, 4): Has a magnitude of 6.4.
- Vector (5, 2): Has a direction of 21.8 degrees.
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