How can you know which one is bigger (sin 130 or sin 50) without using a calculator?

Question

Please help me calculate such values without using a calculator.

Answer 1

To know which one is bigger, consider where they are on the unit circle. The sine corresponds to the y coordinate of the angle when plotted on the unit circle.
http://www.hyper-ad.com/tutoring/math/trig/Trigonometric%20Functions%20of%20Common%20Angles.html
It so happens that 130 degrees has a reference angle of 50 degrees, so it's the same size angle, so it has the same y coordinate, so the sines of both angles should be the same.
For some easy angles, that I call "landmark angles", you can calculate the trig functions by comparing parts of a right triangle laid out with the angle under consideration. This is a common exercise in trig. See for example
http://www.hyper-ad.com/tutoring/math/trig/Trigonometric%20Functions%20of%20Common%20Angles.html
To calculate general trig functiions without using a calculator, you could lay out a triangle the same way, and measrure the parts, but your calculation would be only as good as your picture. A more accurate way would be to use an approximation such as power series or Taylor's series,
http://www.euclideanspace.com/maths/geometry/trig/series/index.htm
(note: to use something like tihs you would first have to convert your angle to radians)
or you'd use the old fashioned methods of slide rule or table of trig values such as the one found at
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/simpleTrig.html

Answer 2

sin x = 1 which is a maximum when x = 90°

But

|130 - 90| = 40
|50 - 90| = 40

Since they are equally far from 90° the values are the same.

Answer 3

All angles are in degrees.

Notice that 130 is a Quadrant 2 angle.
So its reference angle is 180 - 130 = 50.
Now in Quadrant II, the sine function is positive.
So, sin 130 = sin 50.

In fact, you can use the identity: sin(180 - θ) = sin (θ)
So, sin(130) = sin(180 - 50) = sin (50).

Answer 4

Since sin 130 = sin (90 + 40) = sin (90 - 40), they are identical. Remember that sin reflects at 90 degrees.

Answer 5

You can calculate them with a Taylor's series,
but recall that sin(180-x) = sin 180 cos x -cos 180 sin x
= sin x.
So sin 130 = sin 50 (assuming, of couse, that
both angles are in degrees).

Answer 6

um first of all i assume you mean 130 degrees??? ....if you look at a pic of the unit circle... like draw a xy coordinate system with a circle. then draw in a line at 50 degrees and 130 degrees (starting at the positive x axis)...sine is the y coordinate and cosine is the x coordinate....so whichever meets the circle at a higher y coordinate has a larger sine.

Answer 7

The sine (sin) function is symmetric about 90º: sin(90º)=1. 50º and 130º are both 40º away from 90º (90º-50º=40º; 130º-90º=40º). Therefore the sin of either will be the same.

Answer 8

Use graph paper, a protractor and a ruler. But this is easy.

They are equal.

Answer 9

Just do it in your head