Answer:
A, D, and F
Step-by-step explanation:
First off, we're not given the hypotenuse's length, so let's use the Pythagorean Theorem to find it:
[tex]1^2+3^2=c^2\\1+9=c^2\\10=c^2\\c=\sqrt{10}[/tex]
With that, we can refer to SOH-CAH-TOA to help us find sine, cosine, and tangent:
SOH (Sine = Opposite/Hypotenuse)
[tex]\sin{\theta}=1/\sqrt{10}=\sqrt{10}/10[/tex]
CAH (Cosine = Adjacent/Hypotenuse)
[tex]\cos{\theta}=3/\sqrt{10}=3\sqrt{10}/10[/tex]
TOA (Tangent = Opposite/Adjacent)
[tex]\tan{\theta}=1/3[/tex]
From this, we can see that A matches up with sine, and we can eliminate B and C.
Cosecant, secant, and cotangent are all reciprocals of the three basic trig ratios:
[tex]\csc{\theta}=1/\sin{\theta}=\sqrt{10}\\\sec{\theta}=1/\cos{\theta}=\sqrt{10}/3\\\cot{\theta}=1/\tan{\theta}=3[/tex]
D matches with cosecant, and F matches with cotangent, so the correct trig rations for θ are A, D, and F.
