Chapter 2 Trigonometry

Transcription

1 Chapter 2 Trigonometry

2 1 Chapter 2 Trigonometry Angles in Standard Position Angles in Standard Position Any angle may be viewed as the rotation of a ray about its endpoint. The starting position of the ray is called the and the final position is called the. By convention, a counter-clockwise rotation produces a angle and a clockwise rotation produces a angle. To study angles with respect to the Cartesian coordinate system, we must first agree on a set way to place an angle in the coordinate plane. We say that an angle θ (Theta) is in standard position in the coordinate plane if the following two requirements are satisfied. 90 a) b) y 180 x 0, Example 1: Without measuring, draw each angle, in standard position. a) 175 b) 200 page 83 #2-3(alt. letters)

3 Chapter 2 Trigonometry 2 Reference Angles For every angle in standard position, there is a corresponding acute angle (less than 90 ) formed with the and the. This acute angle is called the. The trigonometric ratios of angles in standard position are the same as for the corresponding reference angle, except for perhaps the. Indicate the reference angle in each of the diagrams below and state the relationship between the reference angle and the angle itself. Example 2: Determine the quadrant and the reference angle for each angle in standard position? a) 215 b) 315 Example 3: Determine the measure of the three other angles in standard position, 0 < θ < 360, that have the reference angle of a) 45 b) 30 page #5, 6-7(alt. letters), 10(a)

4 3 Chapter 2 Trigonometry Special Triangles Right triangles are pretty special in their own right. But there are two extra-special right triangles They are triangles and triangles Triangle 45 This is an isosceles triangle, so the legs must be equal. If we make the legs have a length of one, then by Pythagoras Theorem, the length of the hypotenuse is Thus we can now give the exact values for 45 sin 45 cos 45 tan Triangle 30 If the length of one side of the equilateral triangle was 2, what would the other side lengths be? 60 This allows us to determine the exact values for: sin 30 sin 60 cos 30 cos 60 tan 30 tan 60

5 Chapter 2 Trigonometry 4 Example 4: Determine the exact lengths for each of the following a) b) 6 cm 30 x cm 45 3 cm x cm Example 5: Determine the exact lengths for each of the following a) b) 30 m AB BC 10 i. What is DE? ii. What is the exact vertical distance between A and C?

6 5 Chapter 2 Trigonometry The Trigonometric Ratios of the Quadrantal Angles The quadrantal angles are any rotation angle which is a multiple of 90 (ie. 0, 90, 180, 270, 360, ). Another way of describing them is to say that their terminal arm is on one of the. Example 6: Using the diagram and definition below, complete the table without using your calculator. sinθ opposite hypotneuse y r cosθ adjacent hypotneuse x r tanθ opposite adjacent y x ( 0,r ) θ sin θ cos θ ( r,0) ( r,0) tan θ Why are tan90 and tan 270 undefined? ( 0, r) page 83 #5 & Special Right Triangle Worksheet

7 Trigonometric Ratios of Any Angle Chapter 2 Trigonometry 6 The trigonometric ratios can be defined for any angle of rotation by considering a point on the terminal side. If P ( x, y) is on the terminal side, then by the Pythagorean Theorem, x + y r or r x y. Note that because r is a distance, it will always be a positive number. P(x, y) θ sin θ opposite hypotneuse y r cos θ adjacent hypotneuse x r tan θ opposite adjacent y x These definitions allow you to find the values of the trigonometric ratios regardless of the size of the angle, and are determined by the coordinates of a point on the terminal side of θ. Use your calculator to determine cos(150 ) and sin(150 ). Why is one answer positive and the other negative? The sign of the trigonometric ratios is determined by the quadrant that the terminal side is in. Indicate the sign of the trigonometric ratios in each of the quadrants in the diagram below. Make sure that your calculator is set in degree mode! Quadrant II ( neg, pos ) cosθ sinθ tanθ Quadrant III ( neg, neg ) cosθ sinθ tanθ Quadrant I ( pos, pos ) cosθ sinθ tanθ Quadrant IV ( pos, neg ) cosθ sinθ tanθ

8 7 Chapter 2 Trigonometry This means that if sin θ < 0(sine ratio is positive), then θ must terminate in either quadrant or, or ifcos θ < 0(cosine ratio is positive), then θ must terminate in either quadrant or. Example 1: The point P(5, 3) lies on the terminal side of the angle θ. Determine the exact trigonometric ratios for sin θ, cos θ and tan θ. Determine the θ as well. y x Example 2: Determine the exact value for each of the following: a) cos 120 b) tan 240 page 96 #1-5(a, c), 6

9 Chapter 2 Trigonometry 8 4 Example 3: Given that cos θ and that θ terminates in Quadrant IV, determine the exact values of 7 the other trigonometric ratios and the value of θ to the nearest degree. Example 4: If 3 cosθ and 4 tan θ 7 3, determine the exact value ofsin θ. page #8(a, b, c), 10, 11, 13, 16

10 9 Chapter 2 Trigonometry Example 5: Given of a degree. 17 sinθ where 0 θ 360, determine the value(s) of θ to the nearest tenth 20 How many answers will there be to this question? Why? In which quadrant(s) will the answers lie? Draw a diagram to show the possible locations of the answer. How will the reference angle compare in the diagrams? Determine the reference angle. Use the diagram to determine the possible values of θ. The following questions should be completed without a calculator. Example 6: Solve for 3 sinθ and 0 θ What makes this question different from the previous question? page #7, 9(b, c, d)

11 Chapter 2 Trigonometry 10 The Sine Law The Sine Law c B a In any triangle sina sinb sinc a b c or a b sina sinb c sinc A b C The ratio of the sine of any angle to its opposite side is the same for all three angles in any triangle. Where does the Sine Law come from? See this YouTube Video Example 1: Determine the indicated lengths in the triangles below: y 12 cm x page 108 #1(a), 2(a), 4(c), 5(a, c)

12 11 Chapter 2 Trigonometry Example 2: Determine the size of the missing angles in the diagram: 18 km 8.7 km P When solving for an angle, don t forget to apply inverse sine (sin 1 ) M 162 N Example 3: Solve the triangle KGBwith K 37, G 84 and k 12cm.

13 Chapter 2 Trigonometry 12 Example 4: Pudluk s family and his friend own cabins on the Kalit River in Nunavut. Pudluk and his friend wish to determine the distance from Pudluk s cabin to the store on the edge of town. They know that the distance between their cabins is 1.8 km. Using a transit, they estimate the measures of the angles between their cabins and the store, as shown in the diagram. Determine the distance from Pudluck s cabin to the store, to the nearest tenth of a kilometre. Store page #1(c), 3(a), 12 The Ambiguous Case If you are given two sides and one angle (SSA) it is called the ambiguous case because there are three possibilities: a) there is no triangle that exists with the given information b) there is a unique triangle that contains the given information c) there are two triangles that exist with the given information

14 13 Chapter 2 Trigonometry In triangles where Ais acute No triangle possible Condition Example a< h a< bsina B Exactly one triangle a h a bsina B Or a b B Two possible triangles h< a< b or bsina< a< b B If Ais obtuse and a> b, then there is one solution. If a b then there is no solution.

15 Chapter 2 Trigonometry 14 Example 5: For each of the following, determine whether there is no solution, one solution or two solutions. a) In ABCwith A 128, a 36cm and b 45cm If Ais an acute angle, then b) In ABCwith A 50, a 24cm and b 30cm a b one solution a h one solution a < h no solution h < a < b two solutions If Ais an obtuse angle, then a b no solution a> b one solution Example 6: In ABC, A 34 and b 250cm. Determine the range of values of a for which there is a) one triangle b) No triangle possible

16 15 Chapter 2 Trigonometry Example 7: Solve the triangle ABCwith A 40, a 24cm and b 30cm. page #4(a), 6(a, c), 8(a, c), 11, 13

17 Chapter 2 Trigonometry 16 The Cosine Law The Cosine Law is a rule relating the three sides of a triangle with the cosine of one of the angles. A b C c a B For any triangle, ABC, a b + c 2bccosA Or b a + c 2accosB c 2 a 2 + b 2 2abcosC Where does the Cosine Law come from? See this YouTube Video Example 1: Determine the distance x in the diagram below. 35 km km x page #1(a, c), 4(a, c), 6 Example 2: Determine the size of A. A 14 cm 10 cm C 22 cm B

18 17 Chapter 2 Trigonometry To solve for the angle in cosine law, you must do some algebra or memorize the following formula. A cos 1 b c a 2bc 2 Example 3: Solve the triangle ABCwith B 36, a 14cm and c 21cm. Example 4: An aircraft-tracking station determines the distance from a helicopter to two aircraft as 50 km and 72 km. The angle between those two distances is 49. Determine the distance between the two aircraft. 50 km km x page #2(a, c), 3a, 4(e), 5, 10, 15 FIN! Enjoy the rest of your summer.