Questions and Answers
What does 'Θ' represent in bearing terminology?
In the notation 'N 30° E', what does it indicate?
Which of the following describes the WE line in bearings?
What does 'S 45° W' signify in terms of direction?
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Who is represented by 'O' in bearing notation?
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What is the length of side AC in the right triangle?
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If angle B is 41 degrees and 50 minutes, what is the measure of angle A?
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What is the total length (x) when combining side BC and side CD?
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How is the length of side BC calculated using tangent?
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What is the measure of angle C in the triangle?
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What does tan 40° equal in the triangle ΔABE if BE is known to be 4?
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How is the length EC calculated in triangle ΔBCE?
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In triangle ΔCDE, if the value of x is calculated using the equation sin 60° = x / 3.58, what is x?
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If the value of BE is 3.36 in ΔABE, what is the relationship established by the angle?
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What is the approximate numerical value of x if calculated as x = 3.58 sin 60°?
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What is the total distance traveled by the boat from the starting point?
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What is the bearing from the starting point to the endpoint?
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Which equation correctly represents the relationship used to find θ?
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What is the first direction the boat travels before making the turn?
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What value is used to calculate 'C', the distance from the endpoint to the starting point?
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What is the measure of angle B in triangle ABC?
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What is the length of side c in triangle ABC?
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How far is the ladder from the wall when it is 24 feet long and leaning against it at an angle of 18°10'?
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What trigonometric function relates side a and side b in triangle ABC?
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What is the value of side a in triangle ABC?
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What is the defining characteristic of a right triangle?
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In a right triangle, if one angle is 90 degrees, what can be said about the two other angles?
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Which trigonometric function represents the ratio of the opposite side to the hypotenuse?
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In the equation $tan heta = \frac{opp}{adj}$, what does 'opp' refer to?
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What is the correct relationship expressed by the cosine function?
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Study Notes
Bearings and Navigation
- Bearing Definition: Angle measured clockwise from the North-South line to an object sighted.
- Bearings Notation: Expressed in the format of cardinal directions (e.g., N 30° E, S 45° W).
- Bearings Calculation: Involves solving triangles using trigonometric functions to determine distances and angles.
Example Calculations
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Triangle Properties:
- Right triangle with angles A (18° 10'), B (41° 50'), and C (90°).
- Side lengths: AC = 60.95m, CD = 20m, BC = 54.56m.
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Trigonometric Relationships:
- ( tan(18° 10') = 20/AC ) leads to calculating AC.
- ( tan(41° 50') = BC/60.95 ) provides BC length through angle relations.
Boat Navigation Example
- Boat Journey: Travels N29°15' E for 54 miles, then N60°45' W for 95 miles.
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Final Computation:
- Total distance from starting point: C = 109.3 miles.
- Final bearing calculated as N31°08' W using trigonometric identities.
Angle of Elevation and Depression
- Observer and Object Definition: E for eye level of the observer, O for the object sighted.
- Calculation Tables: Use tangent and cosine relations for different triangles (ΔABE, ΔBCE, ΔCDE) to find side lengths.
Right Triangle ABC
- Triangle Information: C = 90°, C = 57°20', side b = 100.
- Angle B Calculation: B = 90° - 57°20' = 32°40'.
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Side Length Calculation:
- Using cos(C) to find side c.
- Using tan(C) for side a, deriving lengths as c = 185.27 and a = 155.97.
Ladder Problem
- Problem Setup: A 24-foot ladder leaning against a wall.
- Distance Calculation: Using ( sin(18°10') = x/24 ), leads to x = 7.5 feet, determining horizontal distance from wall.
Basic Properties of Right Triangles
- Right Triangle Characteristics: Contains one 90° angle; other angles must be acute and complementary.
- Vertices and Angles: Labeled as A, B, C with opposite sides represented as a, b, c.
Trigonometric Functions
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Definitions:
- ( sin \theta = \frac{opp}{hyp} )
- ( cos \theta = \frac{adj}{hyp} )
- ( tan \theta = \frac{opp}{adj} )
- Importance: Key functions for calculating sides and angles in right triangles.
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Description
Test your understanding of bearings and directions in navigation. This quiz covers the concept of bearings from an observer's point of view, including the notation and angle measurements associated with different bearings. Perfect for students studying navigation or geography!
