2. Pythagorean Theorem:

Transcription

1 Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle is called the, and the remaining two sides are called the. 2. Pythagorean Theorem: 3. (1) θ is an, i.e., 0 0 < θ < 90 0, or 0 < θ < π 2. (2) Place θ in standard position, then the coordinates of the point P are. (3) P is a point on the terminal side of θ that is also on the circle. 1

2 (4) We can express the trigonometric functions of θ as ratios of the sides of a right triangle. sin θ = = csc θ = = cos θ = = sec θ = = tan θ = = cot θ = = 4. Find the exact value of the six trigonometric functions of the angle θ in a right triangle with hypotenuse 5 and adjacent (1) If the sum of two angles are a right angle, we say that these two angle are. (2) Side a adjacent to β and opposite α, side b opposite β and adjacent to α. 2

3 (3) Cofunctions sin α = cos α = tan α = csc α = sec α = cot α = Because of these relations, the functions sine and cosine, tangent and cotangent, and secant and cosecant are called of each other. Recall: sin( π 2 θ) = cos θ, cos(π 2 θ) = sin θ. 6. Theorem: Cofunctions of complementary angles are equal. Example: sin 30 0 = cos 60 0, tan 45 0 = cot 45 0, sec 80 0 = csc Example: Simply the following expressions. (1) sin π 12 cos 5π 12 = (2) sin sin = (3) (tan 20 0 ) (tan 70 0 ) = 8. To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles. For the above right triangle, we have 3

4 8. Example: If b = 2 and α = 40 0 in a right triangle, find a, c, and β. (a 1.68, c 2.61) 9. Example: If a = 3 and b = 2 in a right triangle, find c, α, and β. (α ) 4

5 10. Example: A surveyor can measure the width of a river by setting up a transit at a point C on one side of the river and taking a sighting of a point A on the other side. After turning an angle of 90 0 at C, the surveyor walks a distance of 200 meters to point B. Using the transit at B, the angle β is measured and found to be What is the width of the river? (b 72.79) 5

6 4.2 The laws of sines 1. If none of the angles of a triangle is a right angle, the triangle is called. 2. Notation: We label an oblique triangle so that side a is opposite angle α, b is opposite angle β, and side c is opposite angle γ. 3. Solve an oblique triangle: To find the lengths of its sides and the measurement of its angles,we need the length of one side along with: (1) two angles; (2) one angle and one other side; (3) the other two sides. There are four possibilities: Case 1: One side and two angles are known (ASA or SAA). 6

7 Case 2: Two sides and the angle opposite one of them are known (SSA). Case 3: (SAS). Two sides and the included angle are known Case 4: Three sides are known (SSS). 4. Law of sines: For a triangle with sides a, b, c and opposite angles α, β, γ, respectively, Remark: The law of sines is used to solve triangles for which Case 1 or Case 2 holds. 7

8 5. Example: Solve the triangle: α = 40 0, β = 60 0, a = 4. (b 5.39, c 6.13) 6. Example: Solve the triangle: α = 35 0, β = 15 0, c = 5. (a 3.74, b 1.69) 8

9 7. The ambiguous case: (1) No triangle: if a < h = b sin α. (2) One right triangle: if a = h = b sin α. (3) Two triangle: if a < b and h = b sin α < a. (4) One triangle: if a b. 9

10 8. Example: Solve the triangle: a = 3, b = 2, α = (c 4.24) 10

11 9. Example: Solve the triangle: a = 6, b = 8, α = (c , c ) 11

12 10. Example: Solve the triangle: a = 2, c = 1, γ =

13 4.3 The laws of cosines 1. Case 3: Two sides and the included angle are known (SAS). Case 4: Three side are known (SSS). 2. Law of cosines: For a triangle with sides a, b, c and opposite angles α, β, γ, respectively, 3. Remark (1) Law of cosines: The square of one side of a triangle equal the sum of the squares of the other two side minus twice their product times the cosine of their included angle. (2) Special case: Pythagorean Theorem 13

14 4. Example: Solve the triangle: a = 2, b = 3, γ = (α , b ) 14

15 5. Example: Solve the triangle: a = 4, b = 3, c = 6. (α , β , γ ) 15

16 4.4 Area of a triangle 1. The area A of a triangle is, where is b is the base and h is an altitude drawn to that base. 2. Other formulas: 3. Remark: The area A of a triangle equals one-half the product of two of its sides times the sine of their included angle. 4. Example: Find the area A of the triangle for which a = 8, b = 6, and γ =

17 5. Heron s Formula: The area A of a triangle with sides a, b and c is, where s = 1 2 (a + b + c). 6. Example: Find the area of a triangle whose sides are 4, 5, and 6. 17