What is the exact value of sin (22.5 degrees)?

Question

1. Solve for x: (1/2)^x = 16
2. If an ant is 50 feet away from a tree growing straight up, and the ant has to look up at an angle of 70 degrees to see the top of the tree, how tall is the tree?
3. What is the period of y = 3cos ( 4x + π/2) + 5?
4. What are the possible rational roots of f(x) = x⁴ -7x + 8?
5. What is the exact value of sin (22.5 degrees)?

Answer 1

1)

 

To solve for x, we want to get both sides of the equation to have equal bases.

 

(1 / 2)^x = 16

 

(2^-1)^x = 2⁴

 

2^-x = 2⁴

 

 

We now have an equation with common bases.  That base is 2. 

 

Equate the exponents.

 

-x = 4

 

 x = -4

 

 

2)

 

This is a trigonometric question.  The distance between the ant and the tree is 50 feet (horizontal distance).  The ant looks up at an angle of 70 degrees.  This is the angle between the ground and the ant's projected vision.  The vertical height is how tall the tree is.  Based on this, if we draw a diagram of the ant and tree and label it correctly, we will get a right triangle.  When we have a right triangle with known sides and angles, it allows us to use trigonometry. 

 

Use the tangent function, since we are not given a slant distance.

 

tanθ = (side opposite to angle) / (side adjacent to angle)

 

tanθ = (height of tree) / (distance between ant and tree)

 

 

Plug in the numbers into this formula.

 

tan70 = height of tree / 50

 

 

Multiply both sides of equation by 50.

 

50*tan70 = height of tree

 

137.37 feet = height of tree

 

 

3)

 

The period is how long it takes for the function to complete one cycle.  The period is  2π /4 = (1/2)π.

 

 

 

4)

 

According to the rational root theorem, the possible rational roots are ±1, ±2, ±4, and ±8.

 

 

5)

 

To find the exact value of sin(22.5 degrees), we write this as a half angle.

 

sin(45 / 2) =

 

√[(1 - cos45) / 2] =

 

√[((1 - √(2) / 2) / 2] =

 

√[((2 - √2) / 2) / 2] =

 

√[(2 - √2) / 4]