Given that [tex]$\operatorname{Tan} y=\frac{1}{\sqrt{2}}$[/tex], find the value of [tex]$\sin y+\frac{1}{\cos y}$[/tex] such that [tex]$0 < y < 90^{\circ}$[/tex].
Answer (2)
Given tan y = 2 1 , consider a right triangle with opposite side 1 and adjacent side 2 .
Calculate the hypotenuse using the Pythagorean theorem: h = 3 .
Determine sin y = 3 1 and cos y = 3 2 .
Compute sin y + c o s y 1 = 3 1 + 2 3 .
Simplify the expression: 6 2 3 + 3 6 .
Explanation
Analyze the problem and given data We are given that tan y = 2 1 and 0 < y < 9 0 ∘ . We want to find the value of sin y + c o s y 1 . Since tan y = 2 1 , we can consider a right triangle where the opposite side is 1 and the adjacent side is 2 .
Calculate the hypotenuse Using the Pythagorean theorem, we can find the hypotenuse of the triangle: h = 1 2 + ( 2 ) 2 = 1 + 2 = 3
Calculate sine and cosine Now we can calculate sin y and cos y :
sin y = hypotenuse opposite = 3 1 cos y = hypotenuse adjacent = 3 2
Calculate the reciprocal of cosine Next, we find c o s y 1 :
cos y 1 = 2 3
Compute the sum Now we can compute sin y + c o s y 1 :
sin y + cos y 1 = 3 1 + 2 3
Rationalize the denominators To simplify, we rationalize the denominators: 3 1 = 3 3 2 3 = 2 6
Combine the terms Therefore, sin y + cos y 1 = 3 3 + 2 6 = 6 2 3 + 3 6
State the final answer Thus, the value of sin y + c o s y 1 is 6 2 3 + 3 6 .
Examples
Imagine you're designing a ramp for a skateboard park. The angle of the ramp, y , affects how much horizontal space the ramp takes up and how high it goes. If you know the tangent of the angle y (which relates the height and length of the ramp), you can calculate sin y + c o s y 1 to help determine the materials needed and the overall steepness of the ramp. This ensures the ramp is safe and fun for skateboarders!
To find sin y + c o s y 1 for tan y = 2 1 , we use a right triangle to calculate sin y and cos y . The final value is 6 2 3 + 3 6 .
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