In each part, let θ be an acute angle of a right triangle. Express the remaining five trigonometric functions in terms of a. Sin θ = a/3 find cos tan cot csc sec

Answer 1

Since sin θ = a/3 and sin θ = length of the side opposite of θ/ length of the hypotenuse, we have that

length of the side adjacent to \theta = \sqrt{3^{2} - a^{2}}

= \sqrt{9 - a^{2}}

Using the preceding information and the definitions of cos θ, tan θ, cot θ, csc θ, and sec θ we have the following:

cos \theta = length of the side adjacent to \theta/length of the hypotenuse

= \sqrt{9 - a^{2}}/3

tan \theta = length of the side opposite of \theta/length of the side adjacent to \theta

= a/\sqrt{9 - a^{2}}

cot \theta = 1/tan \theta

= 1/[a/\sqrt{9 - a^{2}}]

= \sqrt{9 - a^{2}}/a

csc \theta = 1/sin \theta

= 1/[a/3]

= 3/a

sec \theta = 1/cos \theta

= 1/[\sqrt{9 - a^{2}}/3]

= 3/\sqrt{9 - a^{2}}