MWA 10 7.5 Solving Triangles
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The document provides instructions on using trigonometric functions to find missing angles or sides of right triangles. It begins with reminders of trigonometric definitions and ratios. Examples are then given to demonstrate finding a missing side using the Pythagorean theorem or a trigonometric ratio, and finding a missing angle using an inverse trigonometric function. Tips are provided on determining which method to use based on the information given in a problem.
MWA 10 7.5 Solving Triangles
- 1. Finding Angles Slide 1 In the past few lessons, you have learned how to use sin, cos and tan to find a length of a side in a right triangle. In this lesson, you will learn how to use sin, cos and tan to find a missing angle. Reminders: • (theta) – usually used to identify an angle (a capital letter can also be used to identify an angle) • adjacent – the side that is adjacent to the angle • opposite – the side that is opposite to the angle • hypotenuse – the side that is directly across from the right angle. • degree mode – remember to set your calculator to degree mode – quick check: sin 30 = 0.5
- 2. Slide 2 cos adj yph sin opp yph tan opp dja soh cah toa
- 3. Slide 3 Find the following trig values to 4 decimal places: sin 30 = sin 120 = cos 30 = cos 120 = tan 30 = tan 120 = -1.73215
- 4. sin 30 = sin 120 = cos 30 = cos 120 = tan 30 = tan 120 = 0.5000 0.8660 0.5774 -0.5000 0.8660 Slide 4 Find the following trig values to 4 decimal places: -1.73215
- 5. Slide 5 In the previous lessons, you calculated the sine, cosine or tangent of an angle. Example: sin 85 = 0.9962 It is also possible to calculate the angle when you are given a value. Example: sin A = 0.7071 A = 45 On your calculator, you will use the inverse sine, cosine or tangent buttons to calculate the angle.
- 6. Slide 6 sin A = 0.7071 On your calculator, • enter 0.7071 • press the 2nd button (or shift) • press the sin button You should get an answer of 44.99 A = 45 Is your calculator in degree mode?
- 7. Slide 7 Steps to calculate the angle: • enter the value • press the 2nd button (or shift) • press the sin-1, cos-1, or tan-1 button Try these: Find A: sin A = 0.9962 You should get an answer of A = 85 Is your calculator in degree mode? Find A: tan A = 28.6363 You should get an answer of A = 88 Find A: cos A = 0.5878 You should get an answer of A = 54
- 8. sin A = 0.91847 sin A = 0.8660 cos A = 0.62159 cos A = -0.7071 tan A = 1.7321 tan A = 0.2679 Slide 8 Using the inverse trig functions, find the measure of the angle (to 1 decimal place):
- 9. A = 66.7 A = 51.6 A = 60 A = 15 A = 135 A = 60sin A = 0.91847 sin A = 0.8660 cos A = 0.62159 cos A = -0.7071 tan A = 1.7321 tan A = 0.2679 Slide 9 Using the inverse trig functions, find the measure of the angle (to 1 decimal place):
- 10. If you need to find a missing side and you know… • at least one side • at least one angle (not the right angle) Use SIN, COS, or TAN Slide 10 Tips for Solving Triangle Problems If you need to find the measure of an angle and you know …. • two or more sides and • the angles are unknown Use the inverse of SIN, COS, or TAN (2nd / Shift button is used before SIN, COS, or TAN) If you need to find a missing side and you know… • two sides Use Pythagorean Theorem a2 + b2 = c2
- 11. Example 1: Find the value of x (to two decimal places). Slide 11 2.71 x 4.29
- 12. Example 1: Find the value of x (to two decimal places). Slide 12 2.71 x 4.29 What do you need to find? What do you know? What will you use to find x?
- 13. Example 1: Find the value of x (to two decimal places). Slide 13 2.71 x 4.29 What do you need to find? What do you know? What will you use to find x? Missing side Two sides Pythagorean Theorem 2 2 2 a b c
- 14. Example 1: Find the value of x (to two decimal places). Slide 14 2.71 x 4.29 What do you need to find? What do you know? What will you use to find x? Missing side Two sides Pythagorean Theorem 2 2 2 a b c Identify the sides. 2 2 2 2.71 4.29 c 2 7.3441 18.4041 c 2 7.3441 18.4041 c a b c 2 25.7482 c 2 25.7482 c 5.0743 5.07 c c Reminder – lengths are always positive so the sign is not required.
- 15. Slide 15 57.73 x 35 Example 2: Find the value of x (to two decimal places).
- 16. Slide 16 What do you need to find? What do you know? What do you want to know? 57.73 x 35 Example 2: Find the value of x (to two decimal places). adjacent opposite hypotenuse What will you use to find x? Label the triangle
- 17. Slide 17 What do you need to find? What do you know? What do you want to know? 57.73 x 35 Example 2: Find the value of x (to two decimal places). Missing side Angle and a side Know HYPotenuse adjacent opposite hypotenuse What will you use to find x? OPPosite sin opp hyp Label the triangle
- 18. Slide 18 What do you need to find? What do you know? What do you want to know? 57.73 x 35 Example 2: Find the value of x (to two decimal places). Missing side Angle and a side Know HYPotenuse adjacent opposite hypotenuse What will you use to find x? OPPosite sin opp hyp Label the triangle sin35 57.73 x 0.5736 57.73 x 0.5736 57. 57.73 57 73 .73 x 33.1139 33.11 x x
- 19. Example 3: Find the value of x (to two decimal places). Slide 19 6.48 10.87 x
- 20. Example 3: Find the value of x (to two decimal places). Slide 20 What do you need to find? What do you know? What will you use to find x? 6.48 10.87 x adjacent opposite hypotenuse Label the triangle
- 21. Example 3: Find the value of x (to two decimal places). Slide 21 What do you need to find? What do you know? What will you use to find x? 6.48 10.87 x adjacent opposite hypotenuse Label the triangle Missing angle Two sides: ADJacent and OPPosite Inverse of tan: tan-1 tan opp adj
- 22. Example 3: Find the value of x (to two decimal places). Slide 22 What do you need to find? What do you know? What will you use to find x? 6.48 10.87 x adjacent opposite hypotenuse Label the triangle Missing angle Two sides: ADJacent and OPPosite Inverse of tan: tan-1 tan opp adj 10.87 tan 6.48 x tan 1.6775x 59.1998 59.20 x x Steps to calculate the angle: • enter the value • press the 2nd button (or shift) • press the tan-1 button
- 23. Example 4: A pirate’s treasure is buried at 100 m directly below the surface of the earth. A hole is to be drilled in order to get at the treasure. Drilling begins directly above the treasure but the drilling is off the vertical line by 5. By how much will the hole miss the treasure when the hole is at the same depth as the treasure? Slide 23
- 24. Example 4: A pirate’s treasure is buried at 100 m directly below the surface of the earth. A hole is to be drilled in order to get at the treasure. Drilling begins directly above the treasure but the drilling is off the vertical line by 5. By how much will the hole miss the treasure when the hole is at the same depth as the treasure? Slide 24 100 m 5° x treasure Draw a diagram
- 25. Example 4: A pirate’s treasure is buried at 100 m directly below the surface of the earth. A hole is to be drilled in order to get at the treasure. Drilling begins directly above the treasure but the drilling is off the vertical line by 5. By how much will the hole miss the treasure when the hole is at the same depth as the treasure? Slide 25 What do you need to find? What do you know? What will you use to find x? Label the triangle 100 m 5° x treasure adjacent opposite hypotenuse
- 26. Example 4: A pirate’s treasure is buried at 100 m directly below the surface of the earth. A hole is to be drilled in order to get at the treasure. Drilling begins directly above the treasure but the drilling is off the vertical line by 5. By how much will the hole miss the treasure when the hole is at the same depth as the treasure? Slide 26 What do you need to find? What do you know? What will you use to find x? Label the triangle 100 m 5° x treasure adjacent opposite hypotenuse Missing side (OPPosite) Angle and ADJacent Inverse of tan: tan-1 tan opp adj
- 27. Example 4: A pirate’s treasure is buried at 100 m directly below the surface of the earth. A hole is to be drilled in order to get at the treasure. Drilling begins directly above the treasure but the drilling is off the vertical line by 5. By how much will the hole miss the treasure when the hole is at the same depth as the treasure? Slide 27 What do you need to find? What do you know? What will you use to find x? Label the triangle 100 m 5° x treasure adjacent opposite hypotenuse Missing side (OPPosite) Angle and ADJacent Inverse of tan: tan-1 tan opp adj tan5 100 x 0.0875 100 x 0.0875 1 100 1 0 00 0 x 8.75x m Answer: The drill will miss the treasure by 8.75m.