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Applications of Trigonometric Functions - Assignment Questions | MATH 124, Assignments of Trigonometry

Fayetteville State University (FSU)Trigonometry

Material Type: Assignment; Class: College Trigonometry; Subject: Mathematics; University: Fayetteville State University; Term: Unknown 1989;

Typology: Assignments

Pre 2010

Uploaded on 08/01/2009

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Chapter 4 Applications of Trigonometric Functions
4.1 Right triangle trigonometry; Applications
1. A triangle in which one angle is a right angle (900) 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) θis an , i.e., 00< θ < 900, or 0< θ < π
2.
(2) Place θin standard position, then the coordinates of the point Pare .
(3) Pis a point on the terminal side of θthat is also on the circle .
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Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications

  1. A triangle in which one angle is a right angle ( 900 ) 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., 00 < θ < 900 , 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.

(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 = =

  1. Find the exact value of the six trigonometric functions of the angle θ in a right triangle with hypotenuse 5 and adjacent 3.
  2. (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.
  1. Example: If b = 2 and α = 40^0 in a right triangle, find a, c, and β. (a ≈ 1 .68, c ≈ 2 .61)
  2. Example: If a = 3 and b = 2 in a right triangle, find c, α, and β. (α ≈ 56. 30 )
  1. 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 900 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 200. What is the width of the river? (b ≈ 72 .79)

Case 2: Two sides and the angle opposite one of them are known (SSA).

Case 3: Two sides and the included angle are known (SAS).

Case 4: Three sides are known (SSS).

  1. 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.

  1. Example: Solve the triangle: α = 40^0 , β = 60^0 , a = 4. (b ≈ 5. 39 , c ≈ 6 .13)
  2. Example: Solve the triangle: α = 35^0 , β = 15^0 , c = 5. (a ≈ 3. 74 , b ≈ 1 .69)
  1. Example: Solve the triangle: a = 3, b = 2, α = 40^0. (c ≈ 4 .24)
  1. Example: Solve the triangle: a = 6, b = 8, α = 35^0. (c 1 ≈ 10. 42 , c 2 ≈ 2 .69)

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, respec- tively,
  3. Remark (1) Law of cosines: The square of one side of a triangle equals 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

  1. Example: Solve the triangle: a = 2, b = 3, γ = 60^0. (α ≈ 40. 90 , b ≈ 79. 10 )

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 = 30^0.
  5. Heron’s Formula: The area A of a triangle with sides a, b and c is where s = 12 (a + b + c).
  6. Example: Find the area of a triangle whose sides are 4 , 5 , and 6.