1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ddd [48]
2 years ago
9

1. Derive the half-angle formulas from the double

Mathematics
1 answer:
lilavasa [31]2 years ago
4 0

1) cos (θ / 2) = √[(1 + cos θ) / 2], sin (θ / 2) = √[(1 - cos θ) / 2], tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]

2) (x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).

3) The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°). The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).

4) The <em>linear</em> function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).

<h3>How to apply trigonometry on deriving formulas and transforming points</h3>

1) The following <em>trigonometric</em> formulae are used to derive the <em>half-angle</em> formulas:

sin² θ / 2 + cos² θ / 2 = 1                      (1)

cos θ = cos² (θ / 2) - sin² (θ / 2)           (2)

First, we derive the formula for the sine of a <em>half</em> angle:

cos θ = 2 · cos² (θ / 2) - 1

cos² (θ / 2) = (1 + cos θ) / 2

cos (θ / 2) = √[(1 + cos θ) / 2]

Second, we derive the formula for the cosine of a <em>half</em> angle:

cos θ = 1 - 2 · sin² (θ / 2)

2 · sin² (θ / 2) = 1 - cos θ

sin² (θ / 2) = (1 - cos θ) / 2

sin (θ / 2) = √[(1 - cos θ) / 2]

Third, we derive the formula for the tangent of a <em>half</em> angle:

tan (θ / 2) = sin (θ / 2) / cos (θ / 2)

tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]

2) The formulae for the conversion of coordinates in <em>rectangular</em> form to <em>polar</em> form are obtained by <em>trigonometric</em> functions:

(x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).

3) Let be the point (x, y) = (2, 3), the coordinates in <em>polar</em> form are:

r = √(2² + 3²)

r = √13

θ = atan(3 / 2)

θ ≈ 56.309°

The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°).

Let be the point (r, θ) = (4, 30°), the coordinates in <em>rectangular</em> form are:

(x, y) = (4 · cos 30°, 4 · sin 30°)

(x, y) = (2√3, 2)

The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).

4) Let be the <em>linear</em> function y = 5 · x - 8, we proceed to use the following <em>substitution</em> formulas: x = r · cos θ, y = r · sin θ

r · sin θ = 5 · r · cos θ - 8

r · sin θ - 5 · r · cos θ = - 8

r · (sin θ - 5 · cos θ) = - 8

r = - 8 / (sin θ - 5 · cos θ)

The <em>linear</em> function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).

To learn more on trigonometric expressions: brainly.com/question/14746686

#SPJ1

You might be interested in
I need help hurry plz
Scorpion4ik [409]

Answer:

Hey there!

The solution of -14.5<x, shows that x is greater than and not equal to -14.5.

The third graph from the top shows this.

Hope this helps :)

7 0
3 years ago
Read 2 more answers
what is the total number of outcomes from selecting one entree from a choice of nine entrees and one dessert from a choice of th
Mazyrski [523]
27 is the answer to this question
7 0
3 years ago
Write a polynomial function with the real<br> zeros -3, 0, and 3.
suter [353]

f(x)=x(x-3)(x+3)

7 0
1 year ago
Juan makes 75% of the goals he attempted during soccer practice. If he makes 30 goals at practice, how many does he attempt?
m_a_m_a [10]

Answer:

22.5

Step-by-step explanation:

30/4=7.5

7.5 times 3 = 22.5

Mark brainliest plz!

5 0
2 years ago
Geometry i need help unit 10 assignment
Vladimir79 [104]

Answer:

20 senameters 2

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • The graph of part of linear function g is shown on the grid.
    11·1 answer
  • Over the last week the daily low tempatures in degrees Fahrenheit have been -4,6.2,18 1/2,-5.9,21,-1/4 and 1.75.List these numbe
    10·1 answer
  • A circle of red roses is planted in Clarksville's central park. The circle has a diameter of 8 feet. What is the circle's circum
    10·1 answer
  • If a triangle has a corner that is 80° and the other is 40° what is the value of the third corner
    11·1 answer
  • A landscaper needs 348 pounds of plant food. He has 114 pounds in his truck, and another 46 pound at his shop.
    11·1 answer
  • If you are 2.25 feet above water and then dive directly down 3.75 into the water, how deep in the water will you be? Explain.
    9·2 answers
  • Samia buys 15 1/2 yards of fabric to make shirts for his friends he needs 2 5/6 yards of fabric to make each shirt how many shir
    8·1 answer
  • An angle measures 12°. Find its complement. Do not include units in your answer.
    6·2 answers
  • Find the slope of each line (no links I’ll report)
    13·1 answer
  • Please help will give brainiest!!! The pathway of a quarter being thrown off a bridge can be modeled by the equation h=-16t^2+72
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!