1. Grade 10 Academic (MPM2D) Unit 5: Trigonometry Slope and Angle (Elevations & Depressions)
Mr. Choi
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2. Trigonometry is a branch of mathematics that studies the relationship between the measures of the angles and the lengths of the sides in triangles.
Slope and Angle (Elevation & Depression)
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3. Sine (Sin), COSINE (Cos), TANGENT (Tan):
Opposite side
Adjacent side
Hypotenuse side
Slope and Angle (Elevation & Depression)
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4. Slope and Slope Angles
Slope of a line is given by ratio The slope is related to the angle the line makes with the x-axis, which is called the slope-angle.
Parallel lines have equal slopes and equal slope-angles.
Lines with a positive slope have slope-angle between 0o and 90o.
Lines with a negative slope have slope-angle between -90o and 0o. If it is “-“, it can be expressed as an equivalent angle between 90o and 180o using the expression (slope-angle) + 180o.
Slope and Angle (Elevation & Depression)
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5. Angle of Elevation & Angle of Depression
When the line rises above the horizontal, the angle formed is called the angle of inclination, or angle of elevation of the line When the line falls below the horizontal, the angle formed is called the angle of declination, or angle of depression of the line.
Slope and Angle (Elevation & Depression)
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6. Example 1: Find to the nearest degree, the angle of inclination of a ramp with a slope of 1: 12.
TANGENT
Slope and Angle (Elevation & Depression)
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7. Example 2: Find the slope angle for the line with equation
TANGENT
Slope and Angle (Elevation & Depression)
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8. Example 3: Esther starts her mountain hike from a point 600 m above the sea level and finish at another point 5 km away horizontally and is 750 m above sea level. Assuming she follows a straight path, what will be the average slope angle of her route?
Let be slope angle of the route.
Opposite & Adjacent
TANGENT
Therefore the average slope angle of her route is approx. 1.72o
Slope and Angle (Elevation & Depression)
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9. Example 4: Find the height of a flagpole that casts a shadow 18
Example 4: Find the height of a flagpole that casts a shadow 18.6m long when the angle of elevation of the sun is 56.2°.
Let h be height of the flag pole in m. h
Opposite & Adjacent
TANGENT
Therefore the height of the flag pole is approx m.
Slope and Angle (Elevation & Depression)
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10. Example 5: From an airplane 420m above the ground, the angle of depression of the end of the runway is 32°. How far is the airplane from the end of the runway along the ground?
Let x be distance the plane from the end of the runway in m.
Opposite & Adjacent
TANGENT
Alternate angles
Therefore the plane from the end of the runway is approx. 672 m.
Slope and Angle (Elevation & Depression)
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11. Example 6: (Double Tangent Problem) From a point on the ground, a student sights the top and bottom of a 15m flagpole on top of a building. The two angles of elevation are 64.6° and 57.3°. How far is the student from the foot of the building?
Let x be distance between the point and the building in m
Let h be the height of the building in m
Therefore the distance from the point to the building is approx m
Slope and Angle (Elevation & Depression)
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12. Example 6: Double Tangent Problem (Decimal Version) From a point on the ground, a student sights the top and bottom of a 15m flagpole on top of a building. The two angles of elevation are 64.6° and 57.3°. How far is the student from the foot of the building?
Let x be distance between the point and the building in m
Let h be the height of the building in m
Therefore the distance from the point to the building is approx m
Slope and Angle (Elevation & Depression)
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13. Homework Work sheet: Text: P. 485 #2, 4 -7 P. 498 #14,15,23,29,30
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Slope and Angle (Elevation & Depression)
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14. End of lesson Slope and Angle (Elevation & Depression)
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