Final answer:
Given that tan(e)=-1, the value of sec(e) is found by using the reciprocal identity of cosine, which results in sec(e) being equal to -√2. This follows from the right triangle relationships and the Pythagorean theorem, where the angle associated with tan(e)=-1 leads to equal but opposite sides, resulting in a hypotenuse of √2 and a sec(e) value of -√2.
Step-by-step explanation:
The question asks us to find sec(e) given that tan(e)=-1. Remember that the secant function is the reciprocal of the cosine function, and in a right triangle where the tangent value is -1, this implies an angle where the opposite and adjacent sides are equal in magnitude but have opposite signs. According to the Pythagorean theorem, the hypotenuse will then be √2. Therefore, sec(e), being the reciprocal of cosine which is the adjacent over the hypotenuse, will have a value of √2 with a negative sign due to the adjacent side (cosine) being negative in the case of tan(e) = -1. The correct answer is b) ∑√2.
To find sec(e) using the given tan(e) value, start by writing the basic identity of secant in terms of cosine:
Since tan(e) = -1, and tangent is the ratio of sine over cosine (sin(e)/cos(e)), it implies that sin(e) and cos(e) are equal in magnitude but opposite in sign and since we're dealing with a right-angled triangle, we can use the Pythagorean identity:
With equal sides, this becomes 2cos(e)2 = 1, leading to cos(e) = √(1/2) = 1/√2, with the negative sign due to tan(e) being negative in the specific quadrant. Finally, sec(e) = 1/cos(e) gives us sec(e) = ∑√2.