A person is on the outer edge of a carousel with a radius of 20 feet that is rotating counterclockwise around a point that is centered at the origin. What is the exact value of the position of the rider after the carousel rotates 5pi/12 radians?
a. {5[sqrt(2)-sqrt(6)], 5[sqrt(2)+sqrt(6)]}
b. {5[sqrt(2)+sqrt(6)], 5[-sqrt(2)+sqrt(6)]}
c. {5[sqrt(2)+sqrt(6)], 5[sqrt(2)-sqrt(6)]}
d. {5[-sqrt(2)+sqrt(6)], 5[sqrt(2)+sqrt(6)]}
Can someone explain how to do this?
Sum and differences formula quick check:
1. D
2. D
3. B
4. B
5. D
Just took it.
I will switch to degrees, sensing that you may be more familiar with those units
5π/12 radians = 75° = 30° + 45°
draw a unit circle and consider any point (x,y) on that circle.
Construct any right-angled triangle with base x and height y and hypotenuse 1
let the angle at the centre be Ø
then sinØ = y/1 -----> y = sinØ
and cosØ = x/1 ----> x = cosØ
If we make the radius r instead of 1, our point becomes
(rcosØ, rsinØ) or (20cos75°, 20sin75°)
so we need both cos75° and sin75°
recall sin(a+b) = sina cosb + cosa sinb
sin(75) = sin(30+45) = sin30cos45 + cos30sin45
= (1/2)(√2/2) + (√3/2)(√2/2) = (√2 + √6)/4
similarly:
cos(75) = cos(30+45)
= cos30cos45 - sin30sin45
= (√3/2)(√2/2) - (1/2)(√2/2)
= (√6 - √2)/4
so your point (x,y)
= (20(√6-√2/4) , 20((√2 + √6)/4)
= (5√6 - 5√2 , 5√2 + 5√6)
looks like d) is our match
stobotnik is right
those answers still 100% right thx stobotnik
thank you !!!
apply the distributive property (2 sqrt x - 1)(5 + sqrt x)
simplify the radical 2 sqrt x-1 * 5+ sqrt x
To simplify the expression (2√x - 1)(5 + √x), you can use the distributive property.
(2√x - 1)(5 + √x) = 2√x * 5 + 2√x * √x - 1 * 5 - 1 * √x
= 10√x + 2x - 5 - √x
= (10√x - √x) + 2x - 5
= 9√x + 2x - 5.
So, the simplified form of (2√x - 1)(5 + √x) is 9√x + 2x - 5.
combine like term 2 sqrt x-1 * 5+ sqrt x
To combine like terms in the expression (2√x - 1)(5 + √x), we need to distribute the terms and then simplify:
(2√x - 1)(5 + √x) = 2√x * 5 + 2√x * √x - 1 * 5 - 1 * √x
= 10√x + 2x - 5 - √x
To combine like terms, we can add together the coefficients of √x:
10√x + 2x - 5 - √x = (10 - 1)√x + 2x - 5
= 9√x + 2x - 5.
So, the expression (2√x - 1)(5 + √x) simplifies to 9√x + 2x - 5 after combining like terms.