University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

422

Pages:

2
Author:

Luciano Kirby

Question 1 What are the quotient identities for a trigonometric functions? Answer 1 tan θ = sin θ / cos θ cot θ = cos θ / sin θ Answer 2 There are two quotient identities that can be used in right triangle trigonometry. A quotient identity defines the relations for tangent and cotangent in terms of sine and cosine. Question 2 Find the altitude of an isosceles triangle with base 4.24 feet. The vertex angle of the triangle measures 85° Answer Let x be the altitude of the isosceles triangle. Consider half of the triangle which is a right triangle having an angle of 85°/2 = 42.5° whose adjacent side is the altitude x and the side opposite is half the base 4.242 = 2.12 feet. Using tangent ratio, we can write: tan 42.5° = opp/adj = 2.12/x x = 2.12 / tan 42.5° Use a calculator in DEGREE mode: x ~ 2.32 feet Question 3 Prove the identity 1 / (2 csc 2x) = cos²x tan x Choose the sequence of steps below that verifies the identity A) cos²x tan x = cos²x * (sin x / cos x) = cos x sin x = (sin 2x) / 2 = 1 / (2 csc 2x) B) cos²x tan x = cos²x * (sin x / cos x) = cos x sin x - 1 / (2 csc 2x) C) cos²x tan x = cos²x * (sin x / cos x) = cos x sin x = 2 sin 2x = 1 / (2 csc 2x) D) cos²x tan x = cos²x * (sin x / cos x) = 2 sin 2x = 1 / (2 csc 2x) Answer Prove the identity 1 / (2 csc 2x) = cos²x tan x Consider cos²x * tan x = cos²x * (sin x / cos x) = cos x sin x = (1/2) * (2 sinx cosx) = (1/2) sin 2x = 1 / (2 csc 2x) tan A = sinA / cosA 1/cscA = sinA Correct answer is (A) Question 4 Find the exact value of sin 0° Answer The point at 0° is (1,0), and the y value here is 0. The hypotenuse, although measuring 1 unit, doesn't matter, because 0/a = 0 always equals 0, no matter the value of a. Hence, sin 0° = 0. Question 5 If sin x = e^y, 0