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NemiM [27]
3 years ago
13

Did ancient mathematicians use a ruler to determine the values of sine and cosine in a circle?

Mathematics
2 answers:
12345 [234]3 years ago
6 0

They didn’t use a ruler I think they made their own materials from what they had. Also thank you.

Fofino [41]3 years ago
5 0

Answer and Explanation:

No, they did not use a ruler to compute the values of trig functions. They used geometry to prove theorems about trigonometry, then used those results and ordinary computations involving addition, subtraction, multiplication, division, and square roots to determine the entries in trigonometric tables.

Claudius Ptolemy (85 C.E.–165) created the first trig tables that we know of, and he included proofs of the theorems he used. One of the main theorems he used is what is now called Ptolemy’s theorem.

In order to prove his sum and difference forumlas, Ptolemy first proved what we now call Ptolemy’s theorem.

<em>Ptolemy’s theorem</em>. The product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the opposite sides.

A cyclic quadrilateral is a quadrilateral inscribed in a circle as <em>ABCD</em>. Ptolemy’s theorem says:

<em>AC </em>⋅ <em>BD </em>= <em>AB </em>⋅ <em>CD </em>+ <em>AD </em>⋅ <em>BC</em>

<em />

When one of the diagonals (like ) is a diameter of the circle, that gives you a way to determine the trig functions for sum and differences of angles.

Sometimes, the sum and difference formulas for sine and cosine are still called Ptolemy’s formulas.

sin(<em>α</em> + <em>β</em>) = sin <em>α</em> cos <em>β</em> + cos <em>α</em> sin <em>β</em>

cos(<em>α</em> + <em>β</em>) = cos <em>α</em> cos <em>β</em> − sin <em>α</em> sin <em>β</em>

sin(<em>α</em> − <em>β</em>) = sin <em>α</em> cos <em>β</em> − cos <em>α</em> sin <em>β</em>

cos(<em>α</em> − <em>β</em>) = cos <em>α</em> cos <em>β</em> + sin <em>α</em> sin <em>β</em>

Ptolemy started with angles he knew the trig functions for. These included angles in various isosceles triangles: 45°-45°-90°, 60°-60°-60°, and 36°-72°-72°, the last one described by Euclid in Proposition IV.10. That gave him the trig functions for 36°, 60°, and 90°. Using half-angle formulas (mentioned by Hipparchus (190–120 B.C.E.), he could compute trig functions for half of those angles and a quarter of those angles. Then using the sum and difference formulas he computed the rest of the numbers in his trig table.

<em>Hope this helps!</em> :)

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your neighbor spent three times more money on Halloween candy then you your neighbor spent $27 on Halloween candy write an equat
DaniilM [7]
You spent 9 dollars. why? because your neighbor spent 3 times more than you, which means you spent 1/3 of what he spent. so just multiply by 1/3 to get 9.

this can also be modeled by an equation. 3x = 27

divide 3 on both sides to get

x = 9
6 0
3 years ago
One serving of a certain brand of yogurt contains 12 G of fat the low-fat version contains 38.75% less fat per serving how many
Mkey [24]

Answer:

One serving of law-fat yogurt contains 7.35 g of fat.

Step-by-step explanation:

One serving of

  • a certain brand of yogurt contains 12 g of fat
  • the low-fat yogurt contains 38.75% less fat

Let x g be the amount of fat in low-fat yogurt. Then

12 g of fat - 100%

x g of fat - 100% - 38.75% = 61.26%

Write a proportion

\dfrac{12}{x}=\dfrac{100}{61.25}

Cross multuply

100\cdot x=12\cdot 61.25\\ \\100x=735\\ \\x=\dfrac{735}{100}=7.35

One serving of law-fat yogurt contains 7.35 g of fat.

3 0
3 years ago
What is value of x? please help
bija089 [108]

Answer:

x = 27

Step-by-step explanation:

To find the value of x, create a proportion. A proportion is two equal ratios for similar figures. The two triangles are similar because they are the same shape but not the same size. Create ratios between corresponding sides of figures.

\frac{x+8}{10}=\frac{2x-5}{14}

To solve for x, cross multiply:

(x+8)(14) = (10)(2x-5)

14x + 112 = 20x -50

        112 = 6x - 50

        162 = 6x

          27 = x

7 0
3 years ago
Which equations represent the function?
hammer [34]
The answer would be D
7 0
2 years ago
Timothy draws three isosceles triangles. In each figure, he measures a pair of angles.
Nookie1986 [14]

Answer: I believe that the answer would be "In an isosceles triangle, two of the angles are congruent."

Step-by-step explanation:

8 0
1 year ago
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