Answer: the solutions for the given trig identities problem are:
(a) sec(θ) = / 3
(b) θ = sec^(-1)(/3)
(c) θ = cos^(-1)(3 / )
(d) θ = cos^(-1)(sqrt(1 - 9/))
Step-by-step explanation: To solve this trigonometry problem, we are given the short leg as 3 and the long leg as . Let's find the values of the trigonometric functions using these given lengths.
(a) To find sec(θ), we need to use the definition of secant:
sec(θ) = hypotenuse / adjacent
In this case, the hypotenuse is the long leg, and the adjacent side is the short leg.
Therefore, sec(θ) = / 3
(b) To solve the equation for θ, we can take the inverse secant of both sides:
sec(θ) = / 3
θ = sec^(-1)(/3)
(c) To solve the equation for θ, we can use the reciprocal identity of secant:
sec(θ) = 1 / cos(θ)
1 / cos(θ) = / 3
cos(θ) = 3 /
θ = cos^(-1)(3 / )
(d) To express θ in terms of , we can use the Pythagorean identity:
sin^2(θ) + cos^2(θ) = 1
Since we know the short leg is 3, we can substitute sin(θ) = 3 / hypotenuse:
(3 / )^2 + cos^2(θ) = 1
9/ + cos^2(θ) = 1
cos^2(θ) = 1 - 9/
cos(θ) = sqrt(1 - 9/)
θ = cos^(-1)(sqrt(1 - 9/))