University:
East Georgia State College
Course:
MATH 1540 | Calculus I
Academic year:

2024
Views:

126

Pages:

4
Author:

somospumasl3o

Question 1 Find the shorter diagonal of a parallelogram, if: 𝑙 = 18 cm , 𝑤 = 26 cm , Δ 𝐴 = 3 9 ∘ l=18cm,w=26cm,ΔA=39 ∘ Answer: Let the shorter diagonal be 𝑥 x. Using the law of cosines: 𝑥 2 = 𝑙 2 + 𝑤 2 − 2 ( 𝑙 ) ( 𝑤 ) cos ⁡ ( 3 9 ∘ ) x 2 =l 2 +w 2 −2(l)(w)cos(39 ∘ ) Substitute the values: 𝑥 2 = 1 8 2 + 2 6 2 − 2 ( 18 ) ( 26 ) cos ⁡ ( 3 9 ∘ ) x 2 =18 2 +26 2 −2(18)(26)cos(39 ∘ ) Calculate: 𝑥 ≈ 18.8 cm x≈18.8cm Question 2 Please can you give me an example of the Law of Sines and Law of Cosines with an answer? Answer: Step 1 Given: Law of Sines and Law of Cosines. Step 2 To determine: Provide examples for each with an answer. Step 3 Solution: Law of Cosines: This is used to calculate the angle (or side length) for triangles other than right triangles. Let the sides of a triangle be 𝑎 , 𝑏 , 𝑐 a,b,c, and the included angle is 𝛼 α. Formula: 𝑐 2 = 𝑎 2 + 𝑏 2 − 2 𝑎 𝑏 cos ⁡ ( 𝛼 ) c 2 =a 2 +b 2 −2abcos(α) Using this, the third side can be calculated. Law of Sines: This relates the lengths of sides to the sines of angles in any triangle: sin ⁡ 𝐴 𝑎 = sin ⁡ 𝐵 𝑏 = sin ⁡ 𝐶 𝑐 a sinA = b sinB = c sinC Where 𝐴 , 𝐵 , 𝐶 A,B,C are angles, and 𝑎 , 𝑏 , 𝑐 a,b,c are opposite sides. Question 3 State the Sine Law formula. Answer: The Sine Law relates the angles and sides of a triangle. Given a triangle: 𝑎 sin ⁡ 𝐴 = 𝑏 sin ⁡ 𝐵 = 𝑐 sin ⁡ 𝐶 sinA a = sinB b = sinC c For angles: sin ⁡ 𝐴 = 𝑎 𝑏 ⋅ sin ⁡ 𝐵 sinA= b a ⋅sinB Question 4 What do all these triangles have in common? a) b) c) Answer: Step 1: All the triangles have the same angles: 4 5 ∘ , 4 5 ∘ , 9 0 ∘ 45 ∘ ,45 ∘ ,90 ∘ . Step 2: This makes them isosceles right triangles. The ratio of their sides is always 1 : 1 : 2 1:1: 2 . Question 5 Find the height of the triangle 𝑔 g to the nearest tenth. Answer: Opposite side = 𝑔 , Hypotenuse = 𝑐 = 8.9 , 𝛽 = 63. 5 ∘ Opposite side=g,Hypotenuse=c=8.9,β=63.5 ∘ Using sin ⁡ 𝛽 = 𝑔 8.9 sinβ= 8.9 g : 𝑔 = 8.9 × sin ⁡ ( 63. 5 ∘ ) g=8.9×sin(63.5 ∘ ) 𝑔 = 8.0 g=8.0