Final answer:
The values of the six trigonometric functions for an angle of 3π/4 on the unit circle are sin: √2/2, cos: -√2/2, tan: -1, csc: √2, sec: -√2, and cot: -1.
Step-by-step explanation:
To find the values of the six trigonometric functions for an angle whose terminal point on the unit circle lies at 3π/4, we first need to understand the unit circle and the coordinates of points on it. At an angle of 3π/4, which is in the second quadrant, both sine and cosine have specific values based on the symmetry of the unit circle. The point at 3π/4 has coordinates (-√2/2, √2/2), since the radius of the unit circle is 1.
Thus, the six trigonometric functions are as follows:
- • Sine (sin): sin(3π/4) = y/r = √2/2
- • Cosine (cos): cos(3π/4) = x/r = -√2/2
- • Tangent (tan): tan(3π/4) = sin/cos = -1
- • Cosecant (csc): csc(3π/4) = 1/sin = √2
- • Secant (sec): sec(3π/4) = 1/cos = -√2
- • Cotangent (cot): cot(3π/4) = 1/tan = -1