Question

Suppose that - π/2 ≤ θ ≤ π/2 and that tan(θ) = 0.1. determine value of sec(θ)

​

Answer 1

`\bf \textit{Pythagorean Identities} \\ \quad \\ sin^2(\theta)+cos^2(\theta)=1 \\ \quad \\ 1+cot^2(\theta)=csc^2(\theta) \\ \quad \\ \boxed{1+tan^2(\theta)=sec^2(\theta)}\\\\ -----------------------------\\\\ \pm √(1+[tan(\theta)]^2)=sec(\theta)\qquad tan(\theta)=0.1 \\\\\\ \pm √(1+[0.1]^2)=sec(\theta)`

now, which is it? the +/-? well, we know the tangent is positive, the interval we have, puts the angle on either, the 1st or 4th quadrant, now on the 4th quadrant, the tangent is negative, positive-x, negative-y
however, on the 1st quadrant, is positive, thus, the angle is in the first quadrant, and thus the secant is positive as well, since positive-x and positive-y, so is the + version


`\bf √(1+[0.1]^2)=sec(\theta)\implies √(1+0.01)=sec(\theta) \\\\\\ √(1.01)=sec(\theta)`