How do you find a numerical value for sin and cos without using technology for example sin60degrees cos150degres?

Answer 1

You need to use special triangles in order to do this. Special triangles are right triangles with specific angles for which we can determine the ratio of the side lengths. This ratio can be represented as a fraction, which could be used to answer a question like this.

For the examples you posted, there is the 30-90-60 degree special right triangle. The ratio of the sides is 1, 2, and square root of 3. The 2 is the side opposite the 90 degrees and is the hypotenuse of the triangle. The 1 is the opposite side of the 30 degrees and the square root of 3 is opposite of the 60 degrees.

To find the sine of 60 degrees, imagine 60 degrees represented on a graph as a triangle. The side opposite of the 60 would be the square root of 3, while the hypotenuse is 2. Sine is opposite over hypotenuse, so the answer to this would be the square root of 3 divided by 2.

For the cosine of 150 degrees, imagine 150 degrees as a line on a graph in the second quadrant. Now form the special triangle with the line and the x axis. You'll have a special triangle with 30 degrees in the corner by the origin, as 180-150=30. You can then find the cosine of this 30 degrees using the ratio of the sides mentioned above.

Always make sure to form the triangle with the x axis. Another common special triangle is the 45-90-45 degrees one. The hypotenuse (opposite the 90 degrees) is square root of 2 and both the other sides are 1.