How do you find Sins and Cosines without and calculator For example sin 120?

Answer 1

Assuming you don't want the answer of "instead of an electronic calculator you can look it up in a table in a book, use a slide rule, use a computer"...

The Sine and Cosine of all acute angles are positive and are the ratios between two of the sides of a right angled triangle where the three sides are:

Hypotenuse - side opposite the right (90°) angle;

Adjacent - the side next to the angle which is not the hypotenuse;

Opposite - the side opposite the angle (the third side of the triangle);

then:

sine = opposite/hypotenuse

cosine = adjacent/hypotenuse

For obtuse angles:

sine angle = sine (180° - angle)

cosine angle = - cosine (180° - angle)

For reflex angles180° to 270°:

sine angle = -sine (angle - 180°)

cosine angle = - cosine (angle - 180°)

For reflex angles270° to 360°:

sine angle = -sine (360° - angle)

cosine angle = cosine (360° - angle)

For some angles, an exact value can be found.

30° and 60°:

By dropping a perpendicular of an equilateral triangle a triangle with angles 30°, 60° and 90° is formed, If the original triangle had sides of 2 units, the right angled triangle has sides 1, 2 and √3 (as found by Pythagoras), thus:

sin 30° = cos 60° = 1/2

sin 60° = cos 30° = √3 / 2

45°:

For an isosceles triangle with a right angle between the equal sides, the side lengths are 1, 1 and √2, and the equal angles are 45° giving:

sin 45° = cos 45° = 1/√2

The actual values for sine and cosine can be calculated from the sum of two infinite series as long as the angle x is measured in radians; to convert degrees to radians:

angle_in_radians = angle° x π / 180°

However, for practical purposes only the first few terms of each series need be used as further terms would not affect the value to the required accuracy:

sin (x) = x - x³/3! + x^5/5! - x^7/7! + ... = Σ (-1)^r x^(2r+1)/(2r+1)! for r = 0,1, 2, ...

cos(x) = 1 - x²/2! + x^4/4! - x^6/6! + ... = Σ (-1)^r x^(2r)/(2r)! for r = 0,1, 2, ...

where n! means factorial n, ie n x (n-1) x (n-2) x ... x 2 x 1.

eg 3! = 3 x 2 x 1 = 6

eg 5! = 5 x 4 x 3 x 2 x 1 = 120