What are the sine, cosine, and tangent of 3π/4 radians?

To find the sine, cosine, and tangent of 3π/4 radians, we can start by analyzing this angle in relation to the unit circle.

Location on the Unit Circle: The angle 3π/4 radians is found in the second quadrant, where the sine value is positive and the cosine value is negative. This angle corresponds to 135 degrees.

Reference Angle: The reference angle for 3π/4 is π/4 or 45 degrees. In this case, we can utilize the known sine, cosine, and tangent values of π/4.

Sine Calculation: The sine of 3π/4 can be expressed as:

sin(3π/4) = sin(π - π/4) = sin(π/4)

Since sin(π/4) = √2/2, we have:

sin(3π/4) = √2/2

Cosine Calculation: The cosine of 3π/4 is found similarly:

cos(3π/4) = cos(π - π/4) = -cos(π/4)

Since cos(π/4) = √2/2, we can conclude:

cos(3π/4) = -√2/2

Tangent Calculation: Finally, we look at the tangent:

tan(3π/4) = sin(3π/4) / cos(3π/4)

Substituting the values we found:

tan(3π/4) = (√2/2) / (-√2/2) = -1

Summary:

• sin(3π/4) = √2/2
• cos(3π/4) = -√2/2
• tan(3π/4) = -1

In conclusion, for 3π/4 radians, the sine is √2/2, the cosine is -√2/2, and the tangent is -1.