If tan ( θ ) = 8 6 , 0 ≤ θ ≤ π 2 , then sin ( θ ) equals

Question

 equals    

cos(θ) equals    

sec(θ) equals   

Answer 1

I think some division signs are missing from what you typed. Guessing the problem says this:

If tan(θ)=8/6, and 0 <= θ <= Pi/2, then find sin(θ), cos(θ), and sec(θ).

Now, we can simplify the tan(θ)=4/3, and since 0 <= θ <= Pi/2, we can picture the angle θ in a right triangle. Since its tangent is 4/3, and tangent is defined as opposite over adjacent, we say that the side opposite to θ has length 4 and the side adjacent to θ has length 3.

Because the triangle is a right triangle, we can use the Pythagorean theorem to get the length of the hypotenuse since we know the legs of the triangle are 3 and 4: √(3^2+4^2)=√(9+16)=√25=5. Summarizing, θ is an acute angle in a right triangle that is:

1. opposite to side length 4
2. adjacent to side length 3
3. the hypotenuse has side length 5.

Since sine, cosine, and secant are defined as ratios of these quantities, you just have to pick the right pairs of numbers from these side lengths to divide to get your answers.