1. Grade 10 Academic (MPM2D) Unit 5: Trigonometry Applications of the Trigonometric Ratios
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.
Applications of the Trigonometric Ratios
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3. Notation: In any triangle, the vertices are named using capital letters as in the given example. The lengths of the sides are named using lower case letters that match the opposite vertex.
Applications of the Trigonometric Ratios
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4. Sine (Sin), COSINE (Cos), TANGENT (Tan):
Opposite side
Adjacent side
Hypotenuse side
Applications of the Trigonometric Ratios
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5. Solving Right Triangles To solve a right triangle means to find all the unknown sides and unknown angles. It requires that you find the “missing” three pieces of information. As with congruent triangles, if you are given the proper three pieces of information you will draw a unique triangle. The three cases are: Case 1. SIDE-SIDE-SIDE Triangle Case 2. SIDE-ANGLE-SIDE Triangle Case 3. ANGLE-SIDE-ANGLE Triangle   The strategy to solve a right triangle will require: 1. The Pythagorean Theorem 2. The Sum of the Angles in a Triangle Theorem 3. Trigonometry
Applications of the Trigonometric Ratios
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6. Example 1: A snow plow has a 3. 2 m blade set at an angle of 25°
Example 1: A snow plow has a 3.2 m blade set at an angle of 25°. How wide a path will the snow plow clear?
Let x be width of the path in m.
25o
Adjacent & Hypotenuse
COSINE
3.2 m x
Therefore the snow plow will clear a path of approximately 2.9 m.
Applications of the Trigonometric Ratios
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7. Example 2: A guy wire, 20 m long, supports a tower and forms an angle of 57o with the ground. a) At what height to the nearest metre is the guy wire attached to the tower? b) How far from the base of the tower is the guy wire attached to the tower?
Let x be the distance between the end of the wire on the ground to the base of the tower in m
20 m y
Let y be the height of the tower in m
57o
For Height: Find y:
Use:
b) For Distance: Find x:
Use: x
Opposite & Hypotenuse
Adjacent & Hypotenuse
SINE
COSINE
Therefore the distance between the end of the wire on the ground to the base of the tower is approx. 11 m
Therefore the height of the
tower is approximately 17 m.
Applications of the Trigonometric Ratios
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8. Example 3: A group of students are on an outdoor education trip
Example 3: A group of students are on an outdoor education trip. They leave their campsite and travel 240 m before reaching the first checkpoint. They turn, creating a 42o angle with their previous path, and travel another 180 m to get to the second checkpoint. They turn again and travel the shortest possible path back to their campsite. What area of the woods did their triangular route cover? Determine the length of the returning path.
Let h be the height of the triangular route covered in m.
180 m
For Height: Find h:
Use: h
Opposite & Hypotenuse
42o
SINE
For Area of route:
240 m
Therefore the area of the triangular route is approx m2.
Applications of the Trigonometric Ratios
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9. Example 3: A group of students are on an outdoor education trip
Example 3: A group of students are on an outdoor education trip. They leave their campsite and travel 240 m before reaching the first checkpoint. They turn, creating a 42o angle with their previous path, and travel another 180 m to get to the second checkpoint. They turn again and travel the shortest possible path back to their campsite. What area of the woods did their triangular route cover? Determine the length of the returning path.
Let x be the distance between the right angle formed and the campsite in m.
Let y be the distance between the right angle formed and the 1st check point in m.
180 m r
120.4m
42o
Let r be the distance of the returning path in m. y x
Recall Height:
240 m
For x:
For r:
For y:
Use:
Adjacent & Hypotenuse
COSINE
An alternate way (COSINE LAW) will be taught to find r in the next unit!!
Therefore the returning path is approximately m
Applications of the Trigonometric Ratios
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10. Example 4: Two office towers are 50 m apart
Example 4: Two office towers are 50 m apart. From the 14th floor of the shorter tower, the angle of elevation of the top of the other tower is 33°, and the angle of depression of the base is 39°. Find the height of the other tower.
Let h be the height of the taller tower in m. x
Let x be length from eye level to the top of the tower in m.
33o
39o
Let y be length from eye level to the bottom of the tower in m. h y
50m is the side of the two angles
Adjacent
x and y are the sides of the two angles
Opposite
TANGENT
50 m
h = x + y
Therefore the height of the taller tower is approximately 73 m
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Applications of the Trigonometric Ratios

11. Homework
Work sheet: Applications of Trigonometric Ratios Text: Check the website for updates
Applications of the Trigonometric Ratios
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12. End of lesson Applications of the Trigonometric Ratios
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