Sec θ = − 4 √ 15 15 tan θ = − √ 15 15 Explanation: Recall that sin θ = opposite hypotenuse Hence, the side opposite θ in our question measures 1 unit and the hypotenuse measures 4 units. Since we're dealing with right triangles, we can find the side adjacent θ using pythagorean theorem. Let the adjacent side be a . a 2 + 1 2 = 4 2 a 2 + 1 = 16 a 2 = 15 a = √ 15 Now, let's define secant and tangent. sec θ = 1 cos θ = 1 adjacent hypotenuse = hypotenuse adjacent tan θ = sin θ cos θ = opposite hypotenuse adjacent hypotenuse = opposite adjacent Applying these definitions: sec θ = 4 √ 15 = 4 √ 15 15 tan θ = 1 √ 15 = √ 15 15 The last thing left to do is to find the signs of these ratios. We know that we're in quadrant I I , where sine is positive, and all the other ratios are negative. Since secant is related to cosine, it will be negative. So, our final ratios are: sec θ = − 4 √ 15 15 tan θ = − √ 15 15 Hopefully this helps!
