6 trigonometric values of 315 degrees?

Answer 1

To find exact values, you should use the unit circle.

It is good practice, too, to use radians as this is much more helpful in higher levels of math, so, do yourself a favor and learn how to convert degrees into radians and start thinking about the circle in radians instead of degrees.

315 degrees = (7 [Pi])/4 radians

You can use the following conversion to find this: 90 degrees = Pi / 2 radians.

315 / 90 = x / (Pi / 2), solve for x.

Although you can find decimal approximations using a calculator, it is better to find these on the unit circle and start building your knowledge about radians and how they related to each other and the circle. You will soon find out that most of the time you are dealing with a 3, 4, 5 or right triangle.

Turns out that here we are looking at the right triangle in the 4th quadrant of the unit circle. All angles in the 4th quadrant share certain principles. x is positive and y is negative in the 4th quadrant. This means that cos is going to be positive and sin and tan are going to be negative. All right triangles also have similar properties, their sin and cos are going to be equal or differ only by their sign and will be either Sqrt(2)/2 or -Sqrt(2)/2.

Building this intuitive knowledge about the unit circle and the angles it contains will serve you much better than getting decimal approximations.

sin( (7 [Pi])/4 ) = -1/Sqrt(2) = -Sqrt(2)/2

cos( (7 [Pi])/4 ) = 1/Sqrt(2) = Sqrt(2)/2

To find tangent, you need to do a little arithmetic. We know that tan(x) = sin(x) / cos(x), so,

tan( (7 [Pi])/4 ) = sin( (7 [Pi])/4 ) / cos( (7 [Pi])/4 ) = (-Sqrt(2)/2)/(Sqrt(2)/2) = -1

You can also remember that this is a right triangle and tangent is also going to be 1 or -1 depending on the quadrant.

The cossec, sec and cot values are the reciprocals of these values:

cossec( (7 [Pi])/4 ) = 1/sin( (7 [Pi])/4 ) = 1/(-Sqrt(2)/2) = -Sqrt(2)

sec( (7 [Pi])/4 ) = 1/cos( (7 [Pi])/4 ) = 1/(Sqrt(2)/2) = Sqrt(2)

cot( (7 [Pi])/4 ) = 1/tan( (7 [Pi])/4 ) = cos( (7 [Pi])/4 ) / sin( (7 [Pi])/4 ) = -1

Answer 2

sin(315) = -0.7071

cos(315) = 0.7071

tan(315) = -1

cosec(315) = -1.4142

sec(315) = 1.4142

cot(315) = -1

Note:

0.7071 = sqrt(1/2) = 1/sqrt(2)

1.4142 = sqrt(2)