Final answer:
The exact values for sec(pi/4), csc(pi/4), tan(pi/4), and cot(pi/4) are √2, √2, 1, and 1, respectively, utilizing an isosceles right triangle inscribed in a unit circle.
Step-by-step explanation:
If theta = pi/4, we can find the exact trigonometric values based on the unit circle and the properties of an isosceles right triangle, where the two non-hypotenuse sides are equal. Since we know the sides of such a triangle inscribed in a unit circle are both of length 1/√2, and the hypotenuse is 1, we can proceed to find the values as follows:
- Sec(theta) equals the reciprocal of cosine, which is the adjacent over hypotenuse. Thus sec(pi/4) = 1/cos(pi/4) = √2.
- Csc(theta) equals the reciprocal of sine, which is the opposite over hypotenuse. Thus csc(pi/4) = 1/sin(pi/4) = √2.
- Tan(theta) equals the opposite over adjacent, which are equal in this case. Therefore, tan(pi/4) = sin(pi/4)/cos(pi/4) = 1.
- Cot(theta) is the reciprocal of tangent. So cot(pi/4) = 1/tan(pi/4) = 1.
The exact values for sec(theta), csc(theta), tan(theta), and cot(theta) are √2, √2, 1, and 1, respectively, when theta is pi/4.