Trigonometry. General Outcome: Develop trigonometric reasoning.

Transcription

1 Math 20-1 Chapter 2 Trigonometry General Outcome: Develop trigonometric reasoning. Specific Outcomes: T1. Demonstrate an understanding of angles in standard position [0 to 360 ]. [R, V] T2. Solve problems, using the three primary trigonometric ratios for angles from 0 to 360 in standard position. [C, ME, PS, R, T, V] [ICT: C6 4.1] T3. Solve problems, using the cosine law and sine law, including the ambiguous case. [C, CN, PS, R, T] [ICT: C6 4.1] Mark Assignments 2.1: Page # (1-7, 9, 12, 14, 17) (11, 13, 15) 2.2: Page # (1-6, 8abc, 16) (7, 9, 11-13, 15, 19) 2.3: Page # 1-5(ac), 6, 8ac, 10-14, : Page # 1-2(ac), 4bdf, 7, 8, 10-12, 19, 20 Quiz 2 Date: Chapter 2 Test Date:

2 2.1 Angles in Standard Position In geometry, an angle is formed by two rays with a common. In trigonometry, angles are often interpreted as of a ray. The starting position and final position are called the and of the angle. If the angle of rotation is the angle is (and vise versa). On a Cartesian plane, you can generate an angle by rotating a ray about the origin. The starting position of the ray, along the is the of the angle. The final position, after a rotation about the origin, is the of the angle. An angle is AN ANGLE IN STANDARD POSITION if: - and -. Angles in standard position are ALWAYS shown on the Cartesian plane. The x-axis and y-axis divide the plane into four quadrants.

3 For each angle in standard position, there is a corresponding acute angle called the REFERERENCE ANGLE. The reference angle is. The reference angle is always positive and measures between. Example 1 Sketch an Angle in Standard Position Sketch each angle in standard position. State the quadrant in which the terminal arm lies. a) 36 b) 210 c) 315

4 Example 2 Determine a Reference Angle Determine the reference angle, θ R for each angle θ. Sketch θ in standard position and label the reference angle θ R. a) Θ = 130 b) Θ = 300 Example 3 Determine the Angle in Standard Position Determine the angle in standard position when an angle of 40 is reflected a) in the y-axis b) in the x-axis c) in the y-axis and then in the x-axis 2.1 Angles in Standard Position (Part B) Review of Trigonometric Ratios: There is a relationship between the sides and angles in EVERY right triangle: sinθ = cosθ = tanθ =

5 Special Right Triangles: For angles of 30⁰, 45⁰, and 60⁰, you can determine the exact values of trigonometric ratios. Examples: HERE ARE THE TWO TRIANGLES WITH EXACT VALUES ****************MEMORIZE THESE!!!!!*************** Example 1 Copy and complete the table without using a calculator. Express each ratio using exact values. θ sinθ cosθ tanθ 30⁰ 45⁰ 60⁰

6 Example 2 Find an Exact Distance Allie is learning to play the piano. Her teacher uses a metronome to help her keep time. The pendulum arm of the metronome is 10cm long. For one particular tempo, the setting results in the arm moving back and forth from a start position of 60⁰ to 120⁰. What horizontal distance does the tip of the arm move in one beat? Give an exact answer. You Try The tempo is adjusted so that the arm of the metronome swings from 45⁰ to 135⁰. What exact horizontal distance does the tip of the arm travel in one beat?

7 2.2 Trigonometric Ratios of Any Angle Trigonometric Ratios on the Cartesian Plane: Trigonometric ratios can be used to determine ANY ANGLE using a reference triangle.

8 Example 1 The point P(-8, 15) lies on the terminal arm of an angle, θ, in standard position. Determine the exact trigonometric ratios for sin θ, cos θ, and tan θ. Example 2 Determine the exact value of a) cos135⁰ b) sin240⁰

9 Example 3 Suppose θ is an angle in standard position with terminal arm in quadrant III, and cosθ = 3. What are the exact values of sinθ and tanθ? 4 Example 4 Determine the values of sinθ, cosθ, and tanθ when the terminal arm of quadrantal angle θ coincides with the positive y-axis, θ = 90. You Try! Use the diagram to determine the values of sinθ, cosθ and tanθ for quadrantal angles of 0⁰, 180⁰, and 270⁰. Organize your answers in a table as shown below.

10 KEY IDEAS: Summary: Writing the primary trigonometric ratios in terms of x, y and r. sinθ = cosθ = tanθ = x and y change sign (+/-) depending on which quadrant you are in. r is the length of the terminal arm of the angle and is always positive. Reference Triangles are always drawn using the reference angle! You can use the CAST Rule to verify the sign on your trig ratio. Review 1: The point P(-5, -12) lies on the terminal arm of an angle in standard position. Determine the exact trigonometric ratios for sin θ, cos θ, and tan θ. Review 2: Suppose θ is an angle in standard position with terminal arm in quadrant III, and tanθ = 1. What are the exact values of sinθ and cosθ? 5

11 Quadrantal Angles are angles in standard position whose terminal arms lies on one of the axes. (ie: 0, 90, 180, 270, 360, ) Review 3:Determine the values of sinθ, cosθ, and tanθ for a quadrantal angle of 360. SOLVING FOR ANGLES GIVEN THEIR SINE, COSINE OR TANGENT: 1. Determine which quadrants the solution(s) will be in using the and the. 2. Solve for the. 3. Sketch the reference angle in the appropriate quadrant. Use the diagram to determine the measure of the related angle in standard position. Example 1 Solve an Angle Given Its Exact Sine, Cosine or Tangent Value Solve for θ. a) sinθ = 0.5, 0 θ < 360 b) cosθ = 3 2, 0 θ < 180 c) tanθ = 3, 0 θ < 360

12 Example 2 Given the following trigonometric ratios for θ, where 0 θ < 360, determine the measure of θ, to the nearest tenth of a degree. b) cosθ = b) sinθ = The Sine Law OBLIQUE TRIANGLES: Use this diagram of a scalene triangle in Quadrant I. Use ABC to write an expression for h in terms of b and a y C trigonometric ratio for <A. b h a Use BDC to write an expression for h in terms of a and a trigonometric ratio for <B. A D B x Write an equation that relates the two expressions for h. Any triangle can be positioned with one vertex at the origin, another vertex on the positive x-axis, and the third vertex in Quadrant I or II.

13 The Sine Law You can use the sine law to solve for a side or an angle in any triangle. Example 1 Determine an Unknown Side Length Pudluk s family and his friend own cabins on the Kalit River in Nunavut. Pudluk and has 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 communications tower near the store, as shown in the diagram. a) Determine the distance from Pudluk s cabin to the store, to the nearest tenth of a kilometre. b) Determine the distance from Pudluk s friend s cabin to the store. Example 2 Determine an Unknown Angle Measure a) In PQR, <P = 36⁰, p = 24.8 m, and q = 23.4 m. Determine the measure of <Q, to the nearest degree.

14 b) In LMN, <L = 64⁰, l = 25.2 cm, and m = 16.5 cm. Determine the measure of <N, to the nearest degree. The Ambiguous Case: Sometimes when you are given two sides and an angle to solve, an ambiguous case may occur. An ambiguous case means that. There are three possible outcomes: Check Out Page 105 in the Textbook! The only time you can have the AMBIGUOUS CASE is when you are given. Look back at Examples 1 & 2! Could any of these be an ambiguous case?

15 Example 3 Use the Sine Law in an Ambiguous Case In ABC, <A = 30⁰, a = 24 cm, and b = 42 cm. Determine the measure of the other sides and angles. Round your answers to the nearest unit.

16 2.4 The Cosine Law The Cosine Law The cosine law relates the three side lengths of a triangle to the cosine of one of its Proof for the Cosine Law: angles. Using the Sine & Cosine Laws: When to use the SINE LAW: When to use the COSINE LAW: *Use the sine and cosine laws together to solve any triangle* (find all the missing sides and angles).

17 Example 1 Determine a Distance A surveyor needs to find the length of a swampy area near Fishing Lake, Manitoba. The surveyor sets up her transit at a point A. She measure the distance to one end of the swamp as m, the distance to the opposite end of the swamp as m and the angle of sight between the two as Determine the length of the swampy area, to the nearest tenth of a metre. Example 2 Determine an Angle The Lion s Gate Bridge has been a Vancouver landmark since it opened in It is the longest suspension bridge in Western Canada. The bridge is strengthened by triangular braces. Suppose one brace has a side lengths 14 m, 19 m, and 12.2 m. Determine the measure of the angle opposite the 14-m side, to the nearest degree.

18 Example 3 Solve a Triangle In ABC, a = 11, b = 5, and <C = 20. Sketch a diagram and determine the length of the unknown side and the measures of the unknown angles, to the nearest tenth.