1. Oblique Triangles and Vectors15
Oblique Triangles and Vectors
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Prepared by: Richard Mitchell Humber College

2. 15.1 - Trigonometric Functions of Any Angle

3. 15.1-DEFINITIONS Definition of the Trigonometric Functions
15.1-Trigonometric Functions of Any Angle.
From any point P on the terminal side of the angle, we draw a perpendicular to the x axis forming a right triangle with legs x and y and with a hypotenuse r. The trigonometric ratios are generated as follows.

4. 15.1-DEFINITIONS Definition of the Trigonometric Functions
15.1-Trigonometric Functions of Any Angle.

5. 15.1-DEFINITIONS Definition of the Trigonometric Functions
15.1-Trigonometric Functions of Any Angle.

6. 15.1-DEFINITIONS Definition of the Trigonometric Functions blank
15.1-Trigonometric Functions of Any Angle.
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7. S+ A+ T+ C+ 15.1-DEFINITIONS CAST Rule blank
15.1-Trigonometric Functions of Any Angle.
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8. 15.1-EXAMPLE 1
A point on the terminal side of angle θ has the coordinates (-3, -5). Write the six trigonometric functions of θ to 3 significant digits. S+ A+ θ
15.1-Trigonometric Functions of Any Angle. T+ C+
(-3, -5)
Terminal Side
From the point P (-3, -5) on the terminal side of the angle, we draw a perpendicular to the x axis forming a right triangle with legs x = -3 and y = -5 and with a hypotenuse r = as found by using the Pythagorean Theorem. The six trigonometric ratios are generated as follows.

9. 15.1-EXAMPLES 3 to 6
Use your calculator to verify the following trigonometric ratios.
S A+
T C+
15.1-Trigonometric Functions of Any Angle.
NOTE: There are infinitely many angles that have the exact same value of a trigonometric function. Usually, we only need two positive angles less than 3600 that have the required trigonometric function.

10. 15.1-EXAMPLE 8
Evaluate the following expression to four significant digits.
Note: Use DEGREE MODE
15.1-Trigonometric Functions of Any Angle.

11. 15.1-DEFINITIONS Reference Angles (Examples 9 to 12) S+ A+ blank T+ C+
For an angle in standard position on the coordinate axis, the acute angle that its terminal side makes with the x axis is called the reference angle θ’.
15.1-Trigonometric Functions of Any Angle.
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12. 15.1-EXAMPLE 13
If sin θ = , find two positive values of θ less than 3600 Work to the nearest tenth of a degree.
S A+
T C+
NOTE: According to the CAST Rule, sine is positive ( ) in the first and second quadrants. Thus, our reference angle of is also found in the first and second quadrants.
15.1-Trigonometric Functions of Any Angle.
NOTE: Since our reference angle of is found in the first and second quadrants our two positive values of θ will also be found here. Thus our values are and

13. 15.1-EXAMPLE 14
Find, to the nearest tenth of a degree, the two positive angles less than 3600 that have a tangent of
S A+
T C+
NOTE: According to the CAST Rule, tan is negative (-2,25) in the second and fourth quadrants. Thus, our reference angle of is also found in the second and fourth quadrants.
15.1-Trigonometric Functions of Any Angle.
NOTE: Since our reference angle of is found in the second and fourth quadrants our two positive values of θ will also be found here. Thus our values are and

14. 15.1-DEFINITIONS Special Angles
15.1-Trigonometric Functions of Any Angle.
The angles 0°, 90°, 180°, and 360° are called quadrantal angles because the terminal side of each of them lies along one of the coordinate axes. Notice that the tangent is undefined for 90° and 270°, angles whose terminal side is on the y axis. The reason is that for any point (x, y) on the y axis, the value of x is zero. Since the tangent is equal to y/x, we have division by zero, which is not defined.

15. Law of Sines
15.2-Law of Sines.

16. 15.2-EXAMPLE 18
Solve triangle ABC where A = 32.50, B = and a = 226.
97.80
= 321
15.2-Law of Sines.

17. 15.2-EXAMPLE 18
Solve triangle ABC where A = 32.50, B = and a = 226.
= 417
97.80
= 321
15.2-Law of Sines.

18. 15.2-EXAMPLE 20
Solve triangle ABC where A = 35.20, a = 525 and c = 412.
26.90
15.2-Law of Sines.

19. 15.2-EXAMPLE 20
Solve triangle ABC where A = 35.20, a = 525 and c = 412.
26.90
= 805
117.90
15.2-Law of Sines.

20. 15.2-DEFINITIONS The Ambiguous Case
Another way to check for the ambiguous case is to make a sketch.
15.2-Law of Sines.

21. Law of Cosines
15.3-Law of Cosines.

22. 15.3-STRATEGY Law of Sines vs Law of Cosines
NOTE: We use the law of sines when we have a known side opposite a known angle. We use the law of cosines only when the law of sines does not work, that is, for all other cases. In the figures shown, the heavy lines indicate the known information and might help in choosing the proper law.
15.3-Law of Cosines.

23. 15.3-EXAMPLE 22
Solve triangle ABC where a = 184, b = 125 and C =
38.10
= 92.6
114.70
BLANK
15.3-Law of Cosines.
(Acute Angle)

24. 15.3-EXAMPLE 24
Solve triangle ABC, where a = 128, b = 146 and c = 222.
108.10
33.20
(Largest Angle)
38.70
BLANK
15.3-Law of Cosines.

25. Applications
15.4-Applications.

26. 15.4-EXAMPLE 25 Find the area of the gusset shown below. 33.7
90.00
θ =
15.4-Applications.

27. 15.4-EXAMPLE extra
From a point on level ground between two power poles of the same height, cables are stretched to the top of each pole. One cable is 52.6 m long, the other is 67.5 m long, and the angle of intersection between the two poles is Find the distance between the two poles. θ
15.4-Applications.

28. Addition of Vectors
15.5-Addition of Vectors.

29. 15.5-DEFINITIONS Vector Diagrams (Resultant Vector)
15.5-Addition of Vectors.
If we draw two vectors tip to tail, the resultant R will be the vector that will complete the triangle when drawn from the tail of the first vector to the tip of the second vector. It does not matter whether vector A or vector B is drawn first; The same resultant will be obtained either way.
The parallelogram method will give the same result. To add the same two vectors A and B as before, we first draw the given vectors tail to tail and complete a parallelogram by drawing lines from the tips of the given vectors, parallel to the given vectors. The resultant R is then the diagonal of the parallelogram drawn from the intersections of the tails of the original vectors.

30. 15.5-STRATEGY
Two vectors, A and B, make an angle of with each other as shown in the figure below. If their magnitudes are A = 125 and B = 146, find the magnitude of the resultant R and the angle Ф1 that R makes with vector B.
Φ2 is the angle between the smaller vector A and the resulting vector R determined by using the Law of Sines. Ф2
COMMON VECTOR ANGLE This is the angle that runs between vectors A and B . It is used to calculate Φ3 which is then used for the Law of Cosines. In this example, the common angle is already given.
47.20
132.80 Ф3
15.5-Addition of Vectors.
Φ3 is the angle needed to apply the Law of Cosines and is calculated by subtracting the common vector angle found between vector A and vector B from ( Φ3 = = )
125 Ф1
Φ1 is the angle between the larger vector B and the resulting vector R determined by using the Law of Sines.
146

31. 15.5-EXAMPLE 27
Two vectors, A and B, make an angle of with each other as shown in the figure below. If their magnitudes are A = 125 and B = 146, find the magnitude of the resultant R and the angle Ф1 that R makes with vector B. Ф2
248 Ф3
132.80
125
21.70 Ф1
47.20
146
15.5-Addition of Vectors.

32. 15.5-STRATEGY
Vector A has a magnitude of 125 at an angle of and Vector B has a magnitude of 146 at an angle (Ф4 ) of Find the magnitude of the resultant and the angle (Ф5) that R makes with the x axis.
Y-axis
COMMON VECTOR ANGLE This is the angle that runs between vectors A and B . In this example, the common angle is not given so we must calculate this angle by subtracting the angle that is given between vector B and the x-axis (300) from the angle that is given between vector A and the x-axis (77.20). The resulting angle (47.20) is then used to calculate Φ3 which is used for the Law of Cosines. Ф2
47.20
132.80 Ф3
15.5-Addition of Vectors.
Φ3 is the angle needed to apply the Law of Cosines and is calculated by subtracting the common vector angle found between vector A and B from 1800 ( Φ3 = = )
125 Ф5
Φ5 is the resulting angle that we ultimately want and lies between the resulting vector R and the x-axis. This angle is determined by adding Φ4 (given) and Φ1 which is calculated from the Law of Sines ( Φ5 = = ) Ф1
146 Ф4
Φ4 is given in this example as 300 and is the angle between vector B and the x axis.
X-axis

33. 15.5-EXAMPLE extra
Vector A has a magnitude of 125 at an angle of and Vector B has a magnitude of 146 at an angle (Ф4 ) of Find the magnitude of the resultant and the angle (Ф5) that R makes with the x axis.
248 Ф3
132.80
125 Ф5 Ф1
146 Ф4
15.5-Addition of Vectors.

34. 15.5-STRATEGY Polar to Rectangular Conversions
Vector A has a magnitude of 125 at an angle of and Vector B has a magnitude of 146 at an angle of Find the magnitude of the resultant and the angle that R makes with the x axis.
Polar to Rectangular Conversions
Vector addition and subtraction are best done in Rectangular Form so we convert all of our Polar Coordinates into Rectangular Components using basic trigonometry or directly on our calculator.
15.5-Addition of Vectors.
Add up the total x-components and total y-components. This represents our Resulting Vector in Rectangular Form.
Convert your final answer back into Polar Form using basic trigonometry or directly on your calculator.

35. 15.5-EXAMPLE 28
Find the resultant R of the four vectors shown in Fig (a).
Vector Rx Components Ry Components
A cos 58.0° = sin 58.0° = B cos 148° = sin 148° = C cos 232° = sin 232° = D cos 291° = sin 291° = R Rx = Ry = -18.5
Step 1: Resolve each given vector into their Rx and Ry rectangular components using x = V cosθ and y = V sinθ. Then, add up the total Rx and Ry values giving the resulting vector R in rectangular form.
15.5-Addition of Vectors.

36. 15.5-EXAMPLE 28
Find the resultant R of the four vectors shown in Fig (a).
Vector Rx Components Ry Components
A cos 58.0° = sin 58.0° = B cos 148° = sin 148° = C cos 232° = sin 232° = D cos 291° = sin 291° = R Rx = Ry = -18.5
Horizontal Component
Vertical Component
Step 1: Resolve each given vector into their Rx and Ry rectangular components using x = V cosθ and y = V sinθ. Then, add up the total Rx and Ry values giving the resulting vector R in rectangular form.
15.5-Addition of Vectors.
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Step 2: Convert the Horizontal and Vertical rectangular components R = (-41.5, -18.5) into their equivalent polar form.

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