Ch. 2 Trigonometry Notes

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1 First Name: Last Name: Block: Ch. Trigonometry Notes.0 PRE-REQUISITES: SOLVING RIGHT TRIANGLES.1 ANGLES IN STANDARD POSITION 6 Ch..1 HW: p. 83 #1,, 4, 5, 7, 9, 10, 8. - TRIGONOMETRIC FUNCTIONS OF AN ANGLE IN STANDARD POSITION 8 SINE AND COSINE FUNCTIONS OF AN ANGLE IN STANDARD POSITION: 8 TANGENT FUNCTION OF AN ANGLE IN STANDARD POSITION: 10 RELATING TANGENT WITH SINE AND COSINE: 11 TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES: 1 Ch.. HW: p. 96 # 1 5, THE SINE LAW 16 Ch..3 HW: p. 108 # 1 5 odd letters, 6, 8, 9, 11, 1, THE COSINE LAW 0 Ch..4 HW: p. 119 #1 4 odd letters, #5, 7, 10 1 CH. - REVIEW Ch. Review HW p. 17 # 11 16, 18 4, p130 #1. 7 Created by Ms. Lee 1 of 7

2 .0 Pre-requisites: Solving Right Triangles Recap: SOH CAH TOA Given a right triangle, label each side in reference to the angle, θ. Also, state the sine, cosine, and tangent ratio. Recap: Pythagorean Theorem Obtuse Triangle: A triangle in which one of the angles is an obtuse angle (bigger than 90 ) Acute Triangle: A triangle in which all angles are acute angles (less than 90 ) Created by Ms. Lee of 7

3 Angles in a triangle add to. Angles on a line add to. Examples: Solving a Right Triangle Given Two Sides 1) Solve ABC. Give the measures to the nearest tenth. Solve for AB: Solve for A: Solve for B: Created by Ms. Lee 3 of 7

4 ) Solve ABC. Give the measures to the nearest tenth. Solve for BC: Solve for A: Solve for B: Examples: Solving a Right Triangle Given Two Sides 3) Solve KMN. Give the measures to the nearest tenth. (Given an angle and one side) Solve for KM: Solve for KN: Solve for K: 4) Solve KMN. Give the measures to the nearest tenth. (Given an angle and one side) Solve for KM: Solve for KN: Solve for N: Created by Ms. Lee 4 of 7

5 Homework Questions: 5) Solve ABC. Give the measures to the nearest tenth. Solve for BC: Solve for A: Solve for B: 6) Solve KMN. Give the measures to the nearest tenth. (Given an angle and one side) Solve for KM: Solve for MN: Solve for N: Created by Ms. Lee 5 of 7

6 .1 Angles in Standard Position Angles in Standard Position for 0 θ < 360 : On a Cartesian plane, you can generate an angle by rotating a ray about the origin. The starting position of the ray, along the positive x-axis, is the initial arm of the angle. The final position, after a rotation about the origin, is the terminal arm of the angle. Angle is said to be an angle in standard position if its vertex is at the origin of a coordinate grid and its initial arm coincides with the positive x-axis. Note: When s are formed by rotating counter-clockwise, s are. When s are formed by rotating clockwise, s are. Reference Angles: For each angle in standard position, there is a corresponding acute angle called the reference angle. The reference angle is the acute angle formed between the terminal arm and the x-axis. Draw an angle in standard position, θ = 30. Terminal Arm lies in quadrant What is the reference angle, θ R = Draw an angle in standard position, θ = 140. Terminal Arm lies in quadrant What is the reference angle, θ R = Created by Ms. Lee 6 of 7

7 Draw an angle in standard position, θ = 00. Terminal Arm lies in quadrant What is the reference angle, θ R = Draw an angle in standard position, θ = 30. Terminal Arm lies in quadrant What is the reference angle, θ R = Examples: 1. Determine the measure of the three other angles in standard position, 0 < θ < 360, that have a reference angle of 30. Which of the following angle(s) is(are) NOT in standard position? Created by Ms. Lee 7 of 7

8 Note: Given a circle with radius, r, we can write x and y coordinates in terms of cosθ and sinθ respectively: Ch..1 HW: p. 83 #1,, 4, 5, 7, 9, 10,. - Trigonometric Functions of an Angle in Standard Position Sine and Cosine Functions of an Angle in Standard Position: Given a unit circle, we can write x and y coordinates in terms of cosθ and sinθ respectively: Created by Ms. Lee 8 of 7

9 1. Rotate P around a unit circle counter-clockwise by each given angle θ, and determine x, and y coordinates of P by finding cos θ and sinθ. a) θ = 60 θ = 10 θ 3 = 40 θ 4 = 300 b) θ = 30 θ = 150 θ 3 = 10 θ 4 = 330 Created by Ms. Lee 9 of 7

10 NOTE: As point P rotates around a unit circle, the x, y coordinates change. Notice the sign of the x and y coordinates. Tangent Function of an Angle in Standard Position: The tangent function is named as such because it involves the tangent line to the unit circle at A(1, 0). Examples: Given a diagram of an angle in standard position, determine tangent of the angle. 1) Determine tan 10. ) Determine tan 70 Created by Ms. Lee 10 of 7

11 3) Determine tan 00 4) Determine tan 360 Examples: Given a diagram of θ. Determine tan θ and θ. 5) Determine tan θ and θ. 6) Determine tan θ and θ. 7) Determine tan θ and θ. 8) Determine tan θ and θ. Relating tangent with sine and cosine: Created by Ms. Lee 11 of 7

12 Trigonometric Functions of Special Angles: 45 Angle and its multiples: What s tan( 45 )? 30 Angle and its multiples: 60 Angle and its multiples: What s tan( 30 )? What s tan( 60 )? Created by Ms. Lee 1 of 7

13 Summary: sin θ cos θ tan θ Examples: 1. Determine the exact value of cos ( 10 ). Determine the exact value of cos( 300 ) 3. Determine the exact value of cos( 180 ) 4. Determine the exact value of sin ( 135 ) 5. Determine the exact value of sin( 40 ) Created by Ms. Lee 13 of 7

14 6. Determine the exact value of tan( 10 ). Examples: 1. Solve the equation for 0 θ < a) cosθ = b) cos θ = 3. Solve the equation for 0 θ < 360. a) sin θ = 1 1 b) sin θ = 3. Solve the equation for 0 θ < 360. a) tan θ = 1 b) tan θ = 1 3 Created by Ms. Lee 14 of 7

15 Examples: 1) The point P(-8, 15) is on the terminal arm of an angle θ in standard position. (You try) a. Sketch the angle, θ. b. Determine the distance r from the origin to P. c. Determine the primary trigonometric ratios of θ. d. Determine the measure of θ to the nearest degree. Ch.. HW: p. 96 # 1 5, 6 1 Created by Ms. Lee 15 of 7

16 .3 The Sine Law The Sine Law: The sine law is a relationship between the sides and angles in any triangle. Given a triangle, ABC, where a, b, and c represent the measures of the sides opposite A, B, and C respectively. Then, sin Or a A b B = sin c C = sin sin A sin B = a b sin C = c Proof of the Sine Law: Therefore, Created by Ms. Lee 16 of 7

17 Examples: 1) Determine an unknown side length. The Ambiguous Case: If you are given the measure of two angles and one contained side (ASA), then the triangle is uniquely defined. In other words, there is only ONE way to draw the triangle. However, if you are given two sides and an angle opposite one of those sides (SSA), the ambiguous case may occur. In the ambiguous case, there are three possible outcomes: 1) No such triangle exists that has the given measures; there is no solution ) One triangle exists that has the given measures; there is one solution 3) Two distinct triangles exist that have the given measures; there are two distinct solutions. Examples: Determine the number of triangles that will satisfy the following conditions: Cases: Given an acute angle, A and a < b: 1) In ABC, A = 60, a = 14 cm, and b = 15 cm. Step 1: Sketch a possible diagram. It s easiest to place the known angle at the lower left corner and the angle to be determined at the lower right. This way, the altitude, h, will always be a vertical line. Step : Calculate the altitude. Is there another way to draw the triangle? Created by Ms. Lee 17 of 7

18 ) In ABC, A = 4, a = 6.9 cm, and b = 10.3 cm. 3) In ABC, a = 3 units, b = 6 units, and A = 70. Cases: Given an acute angle, A and a b: 4) In ABC, a = 5 units, b = 3 units, and A = 40. Cases: Given an obtuse angle, A: 5) In ABC, A = 110, a = 11 m, b = 9 m. 6) In ABC, A = 10, a = cm, b = 4 cm. Created by Ms. Lee 18 of 7

19 Pre-Calculus 11 Examples: Solve for the unknown: 7) In LMN, L = 64, l = 5. cm, and m = 16.5 cm. Determine the measure of N, to the nearest degree. Examples: Solve a Triangle: Solving a triangle means to find the measure of all angles and all sides. 8) In ABC, a = 4.8 cm, b = 6.4 cm, and A = 18. Solve the triangle. Round angles to the nearest degree and sides to the nearest tenth of a centimeter. Ch..3 HW: p. 108 # 1 5 odd letters, 6, 8, 9, 11, 1, 4 Created by Ms. Lee 19 of 7

20 .4 The Cosine Law The Cosine Law: The cosine law describes the relationship between the cosine of an angle and the lengths of the three sides of any triangle. For any ABC, where a, b, and c are the lengths of the sides opposite to A, B, and C, respectively, the cosine law states that c = a + b ab cos( C) or a = b + c bc cos( A) b = a + c ac cos( B) Proof of the Cosine Law: Prove: c = a + b ab cos( C) Prove: a = b + c bc cos( A) Created by Ms. Lee 0 of 7

21 Examples: 1. Nina wants to find the distance between two points, A, and B, on opposite sides of a pond. She locates a point C that is 35.5 m from A and 48.8 m from B. If the angle at C is 54, determine the distance AB, to the nearest tenth of a metre.. A triangular brace has side lengths 14 m, 18 m, and m. Determine the measure of the angle opposite the 18-m side, to the nearest degree. 3. In ABC, a = 9, b = 7, and C = Sketch a diagram and determine the length of the unknown side and the measures of the unknown angles, to the nearest tenth. Ch..4 HW: p. 119 #1 4 odd letters, #5, 7, 10 Created by Ms. Lee 1 of 7

22 Ch. - Review 1. In each diagram, determine sinθ and cos θ. Also, determine θ. Roundθ to the nearest degree. a. b. sinθ = cos θ = sinθ = cos θ = θ = θ = c. d. sinθ = cos θ = sinθ = cos θ = θ = θ = e. f. sinθ = cos θ = sinθ = cos θ = θ = θ = Created by Ms. Lee of 7

23 g. h. sinθ = cos θ = sinθ = cos θ = θ = θ =. In each diagram, determine tan θ and θ in degrees (Round to the nearest tenth of a degree). a. b. tanθ = c. θ = tanθ = d. θ = tanθ = θ = tanθ = θ = 3. Determine cosθ. Leave the answers as exact values (Do not round). a. θ = 70 b. θ = 330 c. θ = 135 d. θ = 40 Created by Ms. Lee 3 of 7

24 4. Determine sinθ. Leave the answers as exact values (Do not round). a. θ = 70 b. θ = 180 c. θ = 300 d. θ = Determine tanθ. Leave the answers as exact values (Do not round). a. θ = 45 b. θ = 150 c. θ = 90 d. θ = Solve forθ in degrees. 0 < θ 360 a. sinθ = b. cosθ = c. sinθ = 1 d. cosθ = 1 e. sinθ = 3 f. cosθ = 3 g. sinθ = 0.5 h. cos θ = 0.5 Created by Ms. Lee 4 of 7

25 i. sin θ = 1 j. cos θ = 1 k. sin θ = 0 l. cos θ = 0 7. Solve forθ in degrees. 0 < θ 360 a. tanθ = 3 b. tanθ = 3 3 c. tanθ = 1 d. tanθ = undefined 8. How many triangles are possible from the given measurement? Hint: It s always easiest to draw the triangle with the given angle at the left bottom corner and the angle opposite the other given side at the right bottom corner. Explain your reasoning. [ marks] A = 40, a = 6cm, c = 3. 5cm Created by Ms. Lee 5 of 7

26 9. Solve for (all possible values of) A to the nearest degree. [ marks] 10. Determine the length of the third side to the nearest tenth of a centimeter. [ marks] 11. Determine J to the nearest degree.. [ marks] Created by Ms. Lee 6 of 7

27 1. In ABC, A = 40, a = 7 cm, b = 5cm. How many triangles are possible? Solve the triangle for all cases. Round the values to the nearest tenth. [3 marks] Ch. Review HW p. 17 # 11 16, 18 4, p130 #1 Created by Ms. Lee 7 of 7