Answer each question below. Estimate any irrational answers to two decimal places.
19. If [tex]$x=4 \cos \theta$[/tex] and [tex]$y=4 \sin \theta$[/tex], then [tex]$\sqrt{x^2+y^2}=$[/tex] ?
Answer (1)
Substitute x = 4 cos θ and y = 4 sin θ into x 2 + y 2 .
Simplify the expression to 16 ( cos 2 θ + sin 2 θ ) .
Apply the trigonometric identity cos 2 θ + sin 2 θ = 1 to get 16 ( 1 ) .
Calculate the final value: 16 = 4 , so the answer is 4 .
Explanation
Understanding the Problem We are given that x = 4 cos θ and y = 4 sin θ . We want to find the value of x 2 + y 2 .
Substitution Substitute the given expressions for x and y into the expression x 2 + y 2 : x 2 + y 2 = ( 4 cos θ ) 2 + ( 4 sin θ ) 2
Simplification Simplify the expression: ( 4 cos θ ) 2 + ( 4 sin θ ) 2 = 16 cos 2 θ + 16 sin 2 θ Factor out the 16: 16 ( cos 2 θ + sin 2 θ )
Applying Trigonometric Identity Using the trigonometric identity sin 2 θ + cos 2 θ = 1 , we have: 16 ( cos 2 θ + sin 2 θ ) = 16 ( 1 ) = 16
Final Calculation Finally, we calculate the square root: 16 = 4 Thus, x 2 + y 2 = 4 .
Examples
In physics, if an object moves in a circle with radius 4, its x and y coordinates can be described as x = 4 cos θ and y = 4 sin θ , where θ is the angle. The expression x 2 + y 2 gives the distance of the object from the origin, which is the radius of the circle. This concept is useful in understanding circular motion and oscillations.
