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

2024
Views:

436

Pages:

3
Author:

somospumasl3o

Question 1 Solve each of the following triangles. If there are two triangles, solve both. Round angle measurements to one decimal place and side lengths to two decimal places. 𝑎 = 9 , 𝑏 = 4 , 𝐵 = 2 3 ∘ a=9,b=4,B=23 ∘ 𝑎 = ? a=? 𝑦 = ? y=? 𝐶 = ? C=? Answer: To find angle 𝐴 A: We use the trigonometric identity: sin ⁡ 𝐵 𝑏 = sin ⁡ 𝐴 𝑎 b sinB = a sinA sin ⁡ 2 3 ∘ 4 = sin ⁡ 𝐴 9 4 sin23 ∘ = 9 sinA sin ⁡ 𝐴 = 0.875 ⟹ 𝐴 = 61.5 4 ∘ sinA=0.875⟹A=61.54 ∘ Then: 𝐶 = 18 0 ∘ − 𝐴 − 𝐵 = 18 0 ∘ − 61.5 4 ∘ − 2 3 ∘ = 95.4 6 ∘ C=180 ∘ −A−B=180 ∘ −61.54 ∘ −23 ∘ =95.46 ∘ To determine side 𝑐 c: sin ⁡ 𝐶 𝑐 = sin ⁡ 𝐵 𝑏 c sinC = b sinB 𝑐 = sin ⁡ 2 3 ∘ ⋅ 𝑏 sin ⁡ 95.4 6 ∘ = 10.19 c= sin95.46 ∘ sin23 ∘ ⋅b =10.19 Final Answer: 𝐶 = 95.4 6 ∘ , 𝑐 = 10.19 C=95.46 ∘ ,c=10.19 Question 2 Triangles △ 𝐴 𝐵 𝐶 △ABC and △ 𝐷 𝐸 𝐹 △DEF are right triangles, as shown. △ 𝐴 𝐵 𝐶 ∼ △ 𝐷 𝐸 𝐹 △ABC∼△DEF Which ratios are equal to cos ⁡ ( 𝐵 ) cos(B)? Choose the TWO ratios that apply. Answer: Given △ 𝐴 𝐵 𝐶 △ABC and △ 𝐷 𝐸 𝐹 △DEF: By definition: cos ⁡ 𝐵 = Base Hypotenuse = 𝐵 𝐶 𝐴 𝐵 cosB= Hypotenuse Base = AB BC From similarity property: 𝐵 𝐶 𝐴 𝐵 = 𝐸 𝐹 𝐷 𝐸 AB BC = DE EF The two ratios equal to cos ⁡ 𝐵 cosB are: 𝐵 𝐶 𝐴 𝐵 and 𝐸 𝐹 𝐷 𝐸 . AB BC and DE EF . Question 3 What is trigonometry? Answer: Trigonometry is the study of triangles. It deals with relationships between angles and sides in a triangle. In a right triangle: sin ⁡ 𝜃 = opposite side hypotenuse side cos ⁡ 𝜃 = adjacent side hypotenuse side tan ⁡ 𝜃 = opposite side adjacent side . sinθ= hypotenuse side opposite side cosθ= hypotenuse side adjacent side tanθ= adjacent side opposite side . Question 4 If the measure of an angle is 2 3 ∘ 1 7 ′ 18 " 23 ∘ 17 ′ 18", what is the measure of its supplement? Answer: Two supplementary angles have a sum of 18 0 ∘ 180 ∘ . 18 0 ∘ − 2 3 ∘ 1 7 ′ 18 " = 15 6 ∘ 4 2 ′ 42 " . 180 ∘ −23 ∘ 17 ′ 18"=156 ∘ 42 ′ 42". Question 5 [Triangle] Determine the measure of each interior angle of △ 𝑀 𝐴 𝑁 △MAN and classify the triangle. △ 𝑀 𝐴 𝑁 △MAN is a(n) __________ triangle. Answer: The sum of the measures of a triangle is 18 0 ∘ 180 ∘ : 𝑚 ∠ 𝑀 + 𝑚 ∠ 𝐴 + 𝑚 ∠ 𝑁 = 18 0 ∘ . m∠M+m∠A+m∠N=180 ∘ . 𝑥 + ( 2 𝑥 − 164 ) + ( 𝑥 − 48 ) = 180. x+(2x−164)+(x−48)=180. 4 𝑥 − 212 = 180 ⟹ 𝑥 = 98. 4x−212=180⟹x=98. Thus: 𝑚 ∠ 𝑀 = 9 8 ∘ , 𝑚 ∠ 𝐴 = 3 2 ∘ , 𝑚 ∠ 𝑁 = 5 0 ∘ . m∠M=98 ∘ ,m∠A=32 ∘ ,m∠N=50 ∘ . Since △ 𝑀 𝐴 𝑁 △MAN has an obtuse angle, it is an obtuse triangle.