Discover How to Find Sin and Cos of Any Angle Without a Calculator

In summary, there are various ways to find the sin or cos of an angle without using a calculator. These include using double/triple angle identities, power series, and mental math tricks such as using exact sine ratios of angles to create linear approximations. However, the accuracy of these methods may vary and it ultimately depends on the level of accuracy needed for the specific angle.
  • #1
StephenDoty
265
0
Is there any way to find what sin(x)= or cos(x)= when the angle is not one of the main unit circle angles without using a calculator? Like when you don't have a calculator and need to find the sin or cos of an angle.

Thank you.

Stephen
 
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  • #2
Well there's always double/triple angle identities.

You could also do a power series and add up the first few terms, I guess...
 
  • #3
are there any mental math tricks or a way to visualize what the sin or cos of some angle equals? I mean you can't really carry around a calculator everywhere.

Any help would be appreciated.

Stephen
 
  • #4
Hi Stephen! :smile:

For small angles, sinx = x - x³/6, and cox = 1 - x²/2 + x^4/24, are accurate enough.

For example, it gives sin30º = 0.4997 and cos30º = 0.8661.

Is that close enough? :smile:
 
  • #5
StephenDoty said:
are there any mental math tricks or a way to visualize what the sin or cos of some angle equals? I mean you can't really carry around a calculator everywhere.

Any help would be appreciated.

Stephen

Double and triple identities don't require a calculator, neither does doing an infinite series if the angle is small enough...
 
  • #6
How accurate do you wish your result to be? With basic Trig identities one can make the computation of the sin/cos of any angle into the sin/cos of an angle less than 45 degrees.

After that, either use sin or cos (30 degrees +/- x) expansions to reduce the problem to the sin/cos of an angle less than 20ish degrees, then use sin/cos (15 degrees +/- x) expansions to make the problem angle even smaller. At this point, everything else is still exact, now just approximate your small angle with the power series.

If you don't need it too exact, just a few d.p's, then Remember exact sine ratios of angles 0, 15, 30, 45, 60, 75, 90 degrees and use create a linear approximation from the one closest required angle.
 

1. What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, and tangent, which are abbreviations for the ratios of the sides of a right triangle. These functions are used to find the missing side lengths or angles of a triangle.

2. How do you use the Pythagorean theorem in trigonometry?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In trigonometry, this theorem is used to find the length of a side when given two other side lengths.

3. What is the unit circle and how is it used in trigonometry?

The unit circle is a circle with a radius of 1 and a center at the origin (0,0) in a Cartesian coordinate system. It is used in trigonometry to understand the properties and relationships between the basic trigonometric functions for angles in both degrees and radians.

4. How do you find the trigonometric functions of an angle?

To find the trigonometric functions of an angle, you first need to identify the ratio of the sides of the right triangle formed by that angle. Then, you can use a calculator or trigonometric tables to find the value of the function. Alternatively, you can use the unit circle to find the values for common angles.

5. What is the difference between radians and degrees in trigonometry?

Radians and degrees are two different units for measuring angles. Degrees are based on a circle divided into 360 equal parts, while radians are based on the circumference of a circle divided into 2π equal parts. In trigonometry, radians are often preferred over degrees because they are easier to work with mathematically.

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