How to find the sin of a degree

Question

sin

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Sine expression calculator
Inverse sine calculator
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What is the sin of 40 in degrees?

Cosine calculator ►

Sine expression calculator

Expression with sin(angle deg|rad):

Expression

Result

Inverse sine calculator

sin-1

Degrees

First result

Second result

Radians

First result

Second result

k = ...,-2,-1,0,1,2,...

Arcsin calculator ►

Sine table

x (deg)	x (rad)	sin(x)
-90°	-π/2	-1
-60°	-π/3	-√3/2
-45°	-π/4	-√2/2
-30°	-π/6	-1/2
0°	0	0
30°	π/6	1/2
45°	π/4	√2/2
60°	π/3	√3/2
90°	π/2	1

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This section looks at Sin, Cos and Tan within the field of trigonometry.

A right-angled triangle is a triangle in which one of the angles is a right-angle. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.

In any right angled triangle, for any angle:

The sine of the angle = the length of the opposite side
the length of the hypotenuse

The cosine of the angle = the length of the adjacent side
the length of the hypotenuse

The tangent of the angle = the length of the opposite side
the length of the adjacent side

So in shorthand notation:
sin = o/h cos = a/h tan = o/a
Often remembered by: soh cah toa

Example

Find the length of side x in the diagram below:

The angle is 60 degrees. We are given the hypotenuse and need to find the adjacent side. This formula which connects these three is:
cos(angle) = adjacent / hypotenuse
therefore, cos60 = x / 13
therefore, x = 13 × cos60 = 6.5
therefore the length of side x is 6.5cm.

This video will explain how the formulas work.

The Graphs of Sin, Cos and Tan - (HIGHER TIER)

The following graphs show the value of sinø, cosø and tanø against ø (ø represents an angle). From the sin graph we can see that sinø = 0 when ø = 0 degrees, 180 degrees and 360 degrees.

Note that the graph of tan has asymptotes (lines which the graph gets close to, but never crosses). These are the red lines (they aren't actually part of the graph).

Also notice that the graphs of sin, cos and tan are periodic. This means that they repeat themselves. Therefore sin(ø) = sin(360 + ø), for example.

Notice also the symmetry of the graphs. For example, cos is symmetrical in the y-axis, which means that cosø = cos(-ø). So, for example, cos(30) = cos(-30).
Also, sin x = sin (180 - x) because of the symmetry of sin in the line ø = 90.

For more information on trigonometry click here

Trigonometry is the branch of mathematics concerned with triangles and the relationships between their angles and sides. In fact, in any given right triangle, a function known as “sine,” abbreviated sin, relates the ratio between the opposite side of an angle and the hypotenuse. Using this knowledge of the ratio of the opposite side and the hypotenuse, you can calculate the specific angle in the triangle that produced the two sides.

Determine your angle of interest. In a right triangle, you will find the following three angles: a 90 degree or right angle and two acute angles less than 90 degrees. First decide which acute angle you would like to solve for, as this will determine which side is opposite your angle of interest.

Calculate the measure of each side. Normally you will have at least two sides. You can solve for any missing side by using the Pythagorean Theorem, which states the sum of each leg-squared equals the hypotenuse-squared. For instance, if you had an adjacent of 3 and a hypotenuse of 5, then you would take the square root of 5^2 – 3^2 = sqrt(25 – 9) = sqrt(16) = 4. So your opposite side would be 4.

Divide the measure of the opposite side of your angle by the measure of your hypotenuse. For example, if your opposite side is 4 and your hypotenuse is 5, then divide 4 by 5, giving you 0.8.

Make sure the computed ratio is present on your calculator and hit the sin^-1 key. This “inverse sine” function takes a known ratio and returns the angle that produced that ratio. For example, sin^-1(0.8) = 53.130 degrees. On some calculators, you may have to hit the sin^-1 key first, type in your ratio and then press enter. Either way, once you have your angle, you can figure out the remaining angle by subtracting your result from 90. In the case of a 3-4-5 triangle, you would have 36.870, 53.130 and 90 as your three angles.